(2007-11-02) (MauriceFréchet, 1906) Distance entails a particular topological structure.
Many topological notions (continuity, connectedness, etc.) werefirst introduced in the context of a metric space, where a distance d is defined which is endowed with the following axiomatic properties:
d(x,y) is a nonnegative real number (called distance from x to y).
d(x,y) = d(y,x).
d(x,z) never exceeds d(x,y) + d(y,z) (the triangular inequality ).
d(x,x) = 0
d(x,y) is zero only if x = y (or else, d is called a semidistance).
From a modern viewpoint, topological properties are based on the concept of an open set, as discussed in thenext article. In a metric space, an open set is a set which contains an open ball centeredon every point in it.
(2007-11-02) (Brouwer,1913) Defining a topology is singling out some subsets as open.
Aset E is said to be a topologicalspace when it possesses a specific topology. Formally, a topology is simply a particular collectionof subsets,called open sets verifying the following axiomatic properties (L.E.J. Brouwer, 1913).
The empty set () is open.
The whole set (E) is open.
Any union of open sets is open.
Any intersection of finitely many open sets is open.
In particular, it can be checked that those axioms are verified for the open sets defined as above in the special case ofmetric spaces.
The trivial (or indiscrete or chaotic) topology is { , E } ; only and E are open (in French, this is called topologie grossière ).
At the other extreme, with the discrete topology, every set is open.
A neighborhood of X (X may be eithera point or a set) is a set which contains some open set containing X.
A neighborhood which is an open set... Thus, an open neighborhood of X is just an open set containing X. Open neighborhoods are the only neighborhoods some authors consider. There are good historical reasons for that viewpoint, but the modern nomenclaturehas freed itself from that constraint, so we may speak freely about interestingthings like closed neighborhoods or compact neighborhoods...
The interior Å of a set A is the union of all open setsit contains. (That's the largest open set contained in A.)
A point a of a topological space E is said to be a limit-point of a subset A whenevery [open] neighborhood of a intersects A in at least onepoint besides A. This concept was introduced byCantorin1872 (in the special case of themetric space of real numbers).
(2014-11-26) Generalization of themetric notion of a limit.
An element a of a topological space is said to be the limitof a sequence an of other points when, for any given integer N, there is aneighborhoodof a which contains every an for n ≥ N.
A sequence is said to be convergent if it has a limit. In some coarse topological spaces, a convergent sequence may have more than one limit. However, such spaces are infrequently considered in practice (except, possibly, as a source of counterexamples for honing the definitions of generaltopological concepts). Usually, we only bother with Hausdorff spaces, where the limit of a sequence, if it exists, is necessarily unique.
This concept of a generalized sequence was introduced in 1922 byE. H. Moore (1862-1932) and Herman L. Smith (1892-1950). It used to be called a Moore-Smith sequence. The current name of net was coined in 1955 by John L. Kelley (1916-1999) in his graduate textbook General Topology.
A key motivation for introducing nets is that the counterparts with netsof some specialiazed topological concepts depending on sequences (sequential continuity, sequential compactness, etc.) match thegeneral concepts if nets are substituted for sequences (continuity, compactness, etc.).
(2013-01-04) The open sets are all the unions of sets from a topological base.
There's at least one such basis: the topology itself (the set of all open sets).
The smallest possiblecardinality of a basis isthe weight of the topology.
Subbasis :
A set B of open sets is a subbasis of the topologyif no lesser topology exists where all the elements of B are open sets.
Equivalently, B is a subbasis of the topology if and only if a basis of the topologyis formed by the empty set and all finite intersections of sets of B (including thenullaryintersection, which is equal to the entire space).
Local Basis :
A local basis (or neighborhood basis ) at point x is a family of open setsof which every neighborhood of x contains a member.
A first-countable space is a topological spacefor which there's acountablelocal basis at every point.
Everymetric space is first-countable (the open balls of rational radius centered on x form a countable local basis at point x).
Second-Countable Spaces :
A second-countable space is a topological spacefor which there's a countable basis. A second-countable space is clearly first-countable.
A second-countable space isseparable. It's alsoLindelöf (every open cover contains a countable subcover). In the particular case of metric spaces, the three properties (second-countability, separability and Lindelöf) are equivalent.
A countableproduct of second-countable spaces is second-countable. However an uncountable such product may not even be first-countable.
(2007-11-02) A subset isclosed when its complement isopen
In a topological space (as definedabove) a set is said to be closed when its complement is open (the complement of a subset F of a set E is the subset of E consisting of allthe elements of E which are not in F).
Clearly, a topology on a set E could be specifiedby indicating which of its subsets are closed. If such a viewpoint is adopted, the closed sets must simply verify the following axiomatic properties:
The whole set (E) is closed.
The empty set () is closed.
Any intersection of closed sets is closed.
Any union of finitely many closed sets is closed.
The equivalence of those properties with the axiomatic propertiespreviously stated for open sets is based on the fact that thecomplement of an intersection is the union of the complements,whereas the complement of a union is the intersection of the complements (de Morgan's Laws).
One topology which is best defined this way is the so-called cofinite topology, for which the only closed sets are , E and all its finite subsets. Another example is the Zariski topopogy in algebraic geometry.
The closure (French:adhérence) of a set A is the intersection of all closed setswhich contain it. (It's the smallest closed set containing A.)
The border (or boundary) A of a set A is the intersection of theclosure of A and the complementof theinterior of A. ( A is a closed set, because it's the intersection of two closed sets.)
The complement of a single point in a topological space (such a space can be said to be punctured at that point, if needed). By extension, we may also consider a space punctured at several discrete points (it's the complement of those points). A punctured space can be very different from the whole space topologically (e.g.,punctured torus).
A set is said to be dense in a topological space whenits closure is equal to the entire space.
A set is said to be nowhere dense whenits closure has empty interior.
A subset is said to be almost open if it has theproperty of Baire (i.e., its symmetrical difference with some open set is ameager set).
A topological space is said to be separable if it's theclosure of a countable subset (equivalently, if a countable set is dense in it). For example, the real line is separable because it's the closure of the set of rational numbers , which isindeedcountable.
(2007-12-06) Open sets of F are intersections with F of open sets of E.
A subspace F of a topological space E is a subset of E endowed with the so-called induced topology where an open set of F is defined to be the intersectionwith F of an open set of E.
Equivalently, a closed set of F is defined to be the intersection with F of a closed set of E.
A subspace F is always both open and closed in itself,but it need not be either open or closed in the whole space E. (F could be any subset of E).
(2007-11-09) The fewer the open sets, the coarser the topology.
The basic structure of a topological space is not sufficientto support some general statementsthat require various assumptions about how atopology distinguishes between points. Historically, some of the following "separation axioms" were onceconsidered for inclusion in the general definition of a topological space. Mercifully, none of them have been retained in that general capacity. Each of them just denotes a particular class of topologies which is specifiedwhenever the corresponding properties are needed. The standard characterizations of all separation axioms are of the form:
Two things separated in a weak sense are separated in a stronger sense.
A subscripted T denotes a choice of a separation axiomwithin the most commonly used hierarchy, tabulated below, where Ti implies Tj if i > j.
Most commonly, a topological space is said to be "separated" when isverifies the "Hausdorff condition" (i.e., the separation axiom T2 ). Such a separated topological space is best called aHausdorff space. In a Hausdorff space, a convergent sequence has a unique limit.
Kolmogorov classification, from coarser to finer (T = axiom)
Thetrivialindiscrete or chaotic topology (where every nonempty open set contains everything) is clearly the coarsest possible topology. It doesn't verify any of the above separation axioms (except if the entire space is empty or contain just a single point). At the other extreme, thediscrete topology (where every subset is open) satisfies every conceivable separation axiom. Neither of those two extreme topologiesis very useful, except to provide didactic examples or counterexamples.
On any infinite space, thecofinite topology (where the closedproper subsets are just the finite proper subsets) verifies T1 but nothing stronger. (: The intersection of two nonempty open sets is infinite.)
Any metric spacedoes satisfy all of the above separation axioms.
Proofs (or exposed tautologies) :
T1 => T0 Trivially.
T1 <=> { All finite sets are closed. } : Let F be a finite subset of a T1 space E. Consider a given point y of F. For any other point x of E, there's an open set containing x but not y. The union of all such open sets is an open set containing every point of E except y;it's thus the complement of {y}. Having an open complement, {y} is closed. So is F, as a finite union of closed sets.Conversely, assume that every finite subset of E is closed. Then, for any ordered pair of distinct points (x,y), the complement of {y}is an open set containing x but not y. Therefore, E obeys T1
T2 => T1 : If distinct points x and y have disjoint neighborhoods, we may pick an openset in such a neighborhood of y which doesn't contain x because it doesn'tintersect the relevant neighborhood of x.
T2 <=> { A pair of distinct points is always a disconnected set. } : Consider two distinct points x and y in a space obeying T2 . As they have disjoint neighborhoods, we have two disjoint open sets such that one contains x butnot y and the other contains x but not y. Thus, {x} and {y} aredisconnected from each other. Conversely, assume that two distinct points x and y are always disconnected from eachother. This means that we have two disjoint open sets which contain x and y respectively. As those are disjoint neighborhood of x and y, T2 holds.
(2007-11-09) (Alexandrov & Urysohn, 1923) A set is compact when any open cover includes a finite subcover. (An open cover is a union of open sets containing the prescribed set.)
A set of compact closure is called precompact or relatively compact.
Note that a closed subset of a compact set is compact. Also, the intersection of a closed set and a compact set is compact.
In metric spaces and Hausdorff spaces, all compact sets areclosed, but this need not be so in general topology. A counterexample is the set of integers under the cofinite topology (wherethe nonempty open sets are just complements of finite sets).
: Only finitely many elements are missing fromthe first open set in the cover, so only finitely other open sets from the coverare needed for the rest). This makes any infinite subset compact but not closed.
Historical Origins for Metric Spaces (Fréchet, 1906)
Two closely-related notions are equivalent to compactness in the case of metric or metrizable spaces:
Sequential compactness : A set is sequentially compact when any sequence in it contains a convergent subsequence.
Limit-point compactness : A set is limit-point compact when it contains a limit-point of every infinite subset.
In the Euclidean space n, two famous theorems are equivalent:
Bolzano-Weierstrass theorem (Bolzano, 1817): Every bounded sequence of points has a convergent subsequence.
Heine-Borel theorem : A set is compact iff it's closed and bounded.
A subset of a metric space is called totally bounded when it can be covered by finitelymanyballs of radius r, for any given radius r.
A metric space is compact if and only if it's complete and totally bounded.
Proof (by ChatGPT 3.5) broken down into 3 lemmas:
1/ A compact metric space X is complete : If X wasn't complete, there would be a Cauchy sequence {x_n} in X that does not converge to any point in X. Consider an open cover of X consisting of open balls centered at the elements of that Cauchy sequence. It doesn't include a finite subcover because the sequence does not converge to any point.
2/ A compact metric space X is totally bounded : Assume X is not totally bounded. Then, there exists some r > 0 for which no finite collection of r-balls can cover X. You can construct an open cover {U_n} by considering the r/2-balls centered at the points of X. This open cover has no finite subcover because if it did, you could take the r-ball centered at each point in the finite subcover,and it would cover X, contradicting the assumption.
3/ A complete totally-bounded metric space X is compact : To prove this, consider an open cover {U_i} of X. We want to show that there exists a finite subcover. Since X is totally bounded, there exists a finite subset {x_1, x_2, ..., x_n} of Xsuch that the 1-balls centered at these points cover X. These balls can be denoted as B(x_1, 1), B(x_2, 1), ..., B(x_n, 1). Now, consider each of these balls. Since X is complete, every Cauchy sequence in X converges to a point in X. For each ball B(x_i, 1),construct a Cauchy sequence by selecting points from the open cover {U_i} that intersect this ball. Since these balls cover X, each point in X will belong to at least one such ball. By completeness, each of these Cauchy sequences converges to a point in X. Now, take a finite subcover {U_{i_1}, U_{i_2}, ..., U_{i_k}} from the open cover {U_i}such that each of these chosen open sets corresponds to one of the balls B(x_j, 1) where 1 ≤ j ≤ n.Since each Cauchy sequence converges to a point in X, you can choose, for each i,a point from the Cauchy sequence corresponding to B(x_j, 1) that lies within U_{i}. This gives you a finite subcover.
(2020-10-03) For such a space, every open cover has a locally finite open refinement.
D is said to be a refinement of a cover C when every set of Dis contained in a set of C (it need not be equal to it).
A cover is said to be locally finite when every covered point possessesa neighborhood which intersects only finitely manysets from the cover.
A compact set is necessarily paracompact (because a subcover is a refinement and a finite subcover is locally finite). The converse isn't true.
Every closed subspace of a paracompact space is paracompact. The product of paracompact spaces need not be paracompact, but the product of a compact space and a paracompact space is paracompact.
In a metric space, all subsets of a paracompact set are paracompact.
Nagata-Smirnov Metrization Theorem :
A topological space is said to be metrizable when it's homeomorphic to a metric space. A topological space is said to be locally metrizable when every point had a metrizable open neighborhood.
A locally metrizable space is metrizable iff it's paracompact and Hausdorff. Corollary : A manifold is metrizable if and only if it is paracompact.
(2007-11-17) Spaces in which every point has a compact neighborhood.
As is demonstrated by the Heine-Borel Theorem for metric spaces, compactness and completeness are strongly related but compactnessimplies an overall limitation which is not present in the purely localconcept of completeness.
Traditionally, completeness is only defined for metric spaces (because Cauchy sequences are a purely metrical concept). A loose counterpart of completeness in general topological spaces,must involve some concept of local compactness. All the definitions which have ever been proposed are equivalentto the one featured above in the case ofHaudorff spaces.
Again, local compactness is a relatively minor topological conceptwhich is only loosely relatedto the very important metric concept of completeness (which André Weil extended to uniform spaces in 1937).
Topological Vector Space :
A topologicalvector spaceis locally compactiff it's finite-dimensional.
This classical result is due to Frédéric Riesz (Riesz Frigyes, in Hungarian).
(2012-11-13) A continuous real-valued function defined on a compact setis bounded and attains both of its extreme values.
The original theorem (for functions defined in n-dimensional Euclidean space) was proved in the 1830s by the Germanophone Bohemian mathematician Bernard Bolzano(1781-1848) whose relevant work (Functionenlehre) was only published in1930. That theorem was established independently by Karl Weierstrass (1815-1897) in 1860.
Nowadays, to establish the theorem for compact sets in any topological space (not necessarily a metric or metrizable one) we merely observethat thecontinuous image of a compact set of pointsis a compact set of reals, namely a closed bounded set of real numbers. Thus, there is a minimum and a maximum and both are the images of some points...
A continuous image of a compact set is compact.
Proof : If f iscontinuous andA iscompact, consider any open cover of f (A). Thepreimages of theelements of that cover are open sets (because f is continuous) which cover A. Since A is compact,we can select a finite number of those which cover A. The images of that selection are open by construction and form a finite subcover of the original coverof f (A).
The converse isn't true: Functions that send compacts to compacts aren't necessarilycontinuous. For example, any function that takes on only finitely many values hasthis property (since finite sets are compact) but is not necessarily continuous.
(2012-09-21) The two definitions of a Borel set are usually equivalent.
A -algebra or tribe (the French term tribu was introduced in 1936 by theBourbakistRené de Possel, 1905-1974) is a family of sets closed under countable intersection, countable union and relative complement.
The open sets need not form a tribe. The smallest tribe containing all open sets (the intersection of all tribes that contain all open sets) iscalled the Borel tribe or the Borelian tribe. Its elements are called Borel sets or Borelians.
Borel sets are thus derived from open sets by countable union, countableintersection and relative complement.
Some authors have proposed to define the Borel tribe as the smallest tribe containing all thecompact sets. This definition is usually equivalent to the classicaldefinition presented above, but for some pathological topologies, this ain't so...
(2007-11-02) Characterizing a set by metric properties of the sequences in it.
U is closed if and only ifit contains the limit of all convergent sequences of its own points.
U is compact when any sequence of its points has a subsequencewhich converges in U.
U iscompleteiffany Cauchy sequence of points of U converges in U.
(2007-11-02) A function is continuous iff the inverse image of any open set is open.
Arguably, one of the original motivations of the entire fieldof topology was to characterize continuity in very general terms. This is achieved by the above definition, which looks natural only afteryears of proper mathematical training...
Two equivalent statements characterize a continuous functionf defined over some subset D of one topological space with values in another:
The inverse image of any open set is a set which is open in D.
The inverse image of any closed set is a set which is closed in D.
Recall that a set "open (resp. closed) in D" is the intersection with D of an open set (resp. a closed set).
Continuous functions verify two important properties, respectively known asthe extreme value theorem and the intermediate value theorem (at least, that's the name they have for real functions of a real variable). Namely:
A homeomorphism is a bicontinuous function (which is to say that it'scontinuous andbijective and that its inverse is continuous as well).
An homeomorphism can be construed as an isomorphism of the topological structure. A bijection is an homeomorphism if andonly if it transforms any open set into an open set and any closed setinto a closed set.
Two topological sets are said to be homeomorphic when there's an homeomorphism between them. Two homeomorphic spaceshave identical topological properties.
(2020-01-02) Continuous mapping from the segment ]0,1[ to theopen square ]0,1[ 2
Here, a curve is understood to bea continous function from the open interval into anything. Until Peano proved otherwise in 1890 by describing the first space-filling curve, it was thought that the image (or range) could only be a manifold of dimension 1. The next year (1891) Hilbert improved the aesthetics of Peano's result with a very symmetrical curvefilling the entire (open) unit square. This is now known as the Hilbert curve.
The Hilbert curve is an example of a continuous bijection whose inverse is not continous (at any point).
A continuous bijection whose inverse is also continous is called bicontinuous. A bicontinuous function is also called an homeomorsphism. All topological properties are (by definition) preserved under homeomorphisms. Dimensionality is one of them. A one-dimensional segment isn't homeomorphic to a two-dimensional square, in spite of the existence of the Hilbert curve.
(2012-12-27) A continuous function restricted to a subspace remains continuous.
The restriction of a continuous function to a topological subspace is always continuous. However, there may not alwaysbe a way to extend a continuous function defined on a subspaceto a continous function defined on the whole space...
One example (butcheredby educators with weak topological skills) pertains to the expression f (x) = 1/x (a simplehomographic transformation). This relation does define a continuousbijectionf from D to D when D is one of the following sets. (Continuity is counterintuitive in the first case!)
{} (reals with single unsigned infinity; topology of a circle).
* (the nonzero complex numbers).
{} (projective complex line; topology of a sphere).
However, no continous extension of f can be defined from (all real numbers, including zero) to the closedinterval [] endowed with the usualtopology (where the positive reals do not constituteaneighborhood of negative infinity, and vice-versa).
This is whythree different types of pseudo-numerical infinities are defined (or ought to be defined) in computer algebra systems (CAS):
and : Signed infinities (used mostly in realanalysis).
(2007-11-09) Coarsest topology for which all projections are continuous.
Consider the cartesian product E of finitely or infinitely many sets:
E =
Ei
iI
For each index i, there's a projection function pi which transforms an element x of E into the corresponding component of x in Ei . Formally:
{ x } =
{ pi(x) }
iI
E is best endowed with the leasttopologywhich makes all such projections continuous. (Recall that the "topology" is, formally, the collection ofall open sets.) This so-called product topology can also be described as consisting of all unions (finite or infinite)of finite intersections of sets of the following form:
Ui
where Ui is an open subset of Ei which is different from Ei in only finitely many cases.
iI
If we didn't insist on Ui being aproper open set for onlyfinitely many indices, we would obtain a finer topology known as the box topology.
The above product topology is often called theTychonoff topology (it's the initial topology with respect to the projection maps). It was discovered byAndrei Nikolaevich Tikhonov(1906-1993) in 1926, before he even graduated... Arguably, this is the only "correct" topology to consider over a cartesian product of topological spaces. In particular, it ensures that a map f is continuous if and only if its components fi are continuous.
{ f (x) } =
{ fi(x) }
iI
This desirable theorem would not be true, in general, with the box topology, which is too fine and makes it much harder for a function to be continuous. Similarly, Tikhonov proved that his product topology makesany product ofcompact spaces compact. By comparison, box topology looks like a misguided idea (except for afinite cartesian product, or when almost allcomponents are endowed with thetrivial topology, in whichcases the two concepts coincide).
The cartesian product of any collection ofcompact spaces is compact.
This is one of the most important results of general topology. It helped define the modern concept of compactness based on the Heine-Borel criterion (every open cover has a finite subcover). That definition replaced a definition of compactness, now called sequential compactness, based on the Bolzano-Weierstrass criterion (any sequence has a convergentsubsequence). Both definitions are equivalent for metrizable spacesbut neither implies the other for [some?] other topological spaces.
Tychonoff 's theorem relies on the Axiom of choice. In fact, Tychonoff 's theorem and the Axiom of choice turn out to be equivalent statements.
Cauchy's Mistake (1821) :
Cauchy thought that a function of two variables x and y which is continuous with respect to x and with respect to y must be continuous with respect to (x,y). This ain't so. The following counterexample was produced in 1870 by Johannes Thomae (1840-1921). It has continuous projections but is discontinuous at the (0,0) point. (: On the line y = a x, the function f has a constant value which depends on a.)
f (x,0)
=
0
If y 0 , f (x,y)
=
sin ( 4 Arctg
x
)
y
(2007-11-02) A connected setcannot be split by two disjoint open sets.
By definition :
Two sets are said to be disconnected from each other if they are respectively contained in two disjoint open sets.
A set is said to be disconnected if it's the union of two nonempty parts that are disconnected from each other.
A set is said to be connected if it's not disconnected.
To prove that a set A is connected, we may showthat it can't be contained in the union of of two disjoint open sets U and V unless one is empty.
A nonempty topological space E is connected if and only if it doesn't contain any clopen (i.e., both open and closed) nonemptyproper subset.
The empty set is connected. So is any set containing only one point.
A topological space where there are no other connected sets is said tobe totally disconnected (in such spaces,there are nonconnected sets with more than one point). For example, thediscrete topology alwaysproduces a totally disconnected topological space. More interestingly, the followings spaces are totally disconnected:
On the other hand, with thetrivial topology (the so-called indiscrete topology) every set is connected.
The closure of a connected set is connected.
Proof : If the closure of a set is disconnected,then that closure can be split by two disjoint open sets whichalso split the set, proving it's disconnected.
Connected Components :
A connected component of a topological space isa maximal connected nonempty set (i.e., a nonempty connected set which isn'tcontained in any larger connected set).
The convention is thus made that the empty set isn't a connected component of anything (not even itself) although it's definitely connected.
All the connected components form a unique partition of the topological space (i.e., they're pairwise disjoint andtheir union is the whole space). This fundamental property is the reason why we had to rule outthe empty set as a connected component (since elements of a partition are never empty).
Every connected component must be closed (: its closure is connected). If there are only finitely many of them, each is also open (: its complement is a finite union of closed sets). If there are infinitely many connected components,they're not necessarily open (e.g.,Cantor set).
Connectedness and Continuity :
Theabove definition of continuity satisfies the intuitiverequirement that a continuous function must transform a connected set into a connected set...
A continuous image of a connected set is connected.
Proof : Let f be a continuous function. By definition,it is such that the inverse image of anyopen set is an open set. We have to provethat the direct imagef (A) of a connected set A is connected or, equivalently, the contrapositive statement: If f (A) is disconnected, so is A.
Well, if f (A) is disconnected, we can split it into twononempty parts respectively contained in two disjoint open sets U and V. Consider the open sets f -1(U ) and f -1(V )...
Neither has an empty intersection with A, because U and V both have at least one element from f (A).
Every element of A is in one or the other of those two open sets.
Their intersection is empty,because any element of both would need to have its image in both U and V, which are disjoint.
A is thus split into nonempty parts by two disjoint open sets.
Note, however, that the converse is false: There are discontinuous functions which transform everyconnected set into a connected set. The following section provides many such examples in thespecial case of real functions of a real variable...
(2014-09-01) A continuous function from reals to reals maps an interval to an interval.
On the real line, theconnected sets are theintervals. So, that's justa special case of the general result established in theprevious section (a continuous image of a connected set is connected).
Spelled out in elementary terms, this yields a very useful result whichsays that, for any continuous function of a real variabledefined between a and b, any value y between f (a) and f (b) is equal to f (x), for some x.
A popular formulation used by Bolzano (for functions on an interval) is:Continuous functions with positive and negative values vanish somewhere.
The converse isn't true :
The intermediate-value property is not a characteristic property of continuous functions: There are functions which are not continuous for which the property holds. Such is the case for any discontinuous derivative f ' of a differentiable function. For example, we may use:
f (x) = x 2 sin ( 1/x ) [ with f (0) = 0 ]
Proof : Since any differentiable function f is continuous, the extreme value theorem states that it must reacha minimum and a maximum within any interval [a,b] on which it is defined. If f ' (a) and f ' (b) have opposite (nonzero) signs, at least one ofthose extrema is not located at an extremity, so it must be at a point x where f ' (x) = 0
(2012-12-30) A path-connected set isconnected. The converse may not be true.
A subset Y is said to be path-connected (or pathwise connected ) when such a path exists whose image is contained in Y, for any pair of extremities {a,b} in Y.
A set consisting of a single point is path-connected. The empty set isn't. (Thus the empty set is a trivial example of a connected setwhich isn't path connected.)
A nontrivial example of a connected set which isn't path-connected isthe closure of the so-called topologist's sine curve ; theplanar curve of cartesian equation:
y = sin ( 1/x ) for 0 < x < 2/
That closure includes the segment at x = 0, between (0,-1) and (0,1).
The interval [0,1] is connected (proof by contradiction).
Any path-connected set is connected.
Proof : Consider two arbitrary points a and b of a path-connected set Y. Let P be the image of a path joining them within Y. If the two extremities were respectively in two disjoint opensets U and V whose union contained Y, then those two open sets would likewise split P and prove itto be disconnected. Since we know that P is connected (as acontinuous image of the connectedset [0,1] examined in our lemma) we deduce that a and b cannot possibly be in two disjoint open sets covering Y. As this is true of any pair of points of Y, there cannot be two nonemptyparts of Y in disjoint open sets covering Y. Therefore, Y is connected.
In a normed space,ballsandconvex sets are path-connected. (: Consider the pathf (u) = (1-u)a + ub )
In a normed space,a connected open set is path-connected.
Proof : For any point a of a nonempty open set U, we may define the following two sets V and W. Both are open. (: Open balls are path-connected (lemma). The union of two arcs sharing an extremity is an arc.)
V consists of all points b of U for which there is a pathfrom a to b.
W consists of all points z of U for which there's no path from a to z.
U is the union of the two disjoint open sets V and W. V is nonempty, since it contains a. Therefore, if U is connected, W must be empty, which means that there's a path from a to any other point of U.
An arc is said to join its pair of extremities (defined as theimage of {0,1} under anyhomeomorphism between the arc and the interval [0,1] ).
A topological space where any point is joined to any other point by an arcis said to be arc-connected or arcwise connected.
Clearly, an arc-connected space ispath-connected (since bicontinuous functions are continuous). However, the converse need not be true.
A simple counterexample is a topological space X consistingof just two points under the trivial topology: That space X is not arc-connected because there are no injections from [0,1] to X,because [0,1] has more than two elements (hence no bicontinuous functions between [0,1] and the only pair of points in X). On the other hand, there's a continuous path from one point of X to the other, obtained from a function which is equal to one point of X atzero and the other point elsewhere. (That function is continuous because the inverse image of the only nonemptyopen set of X is equal to [0,1], which is open in itself.)
(2007-10-31) (Jordan, 1866)
A homotopy between two continuous functions f and g from X to Y is a continuous function h from X [0,1] to Y such that
x h (x,0) = f (x) and h (x,1) = g (x)
If such a homotopy exists, thefunctions f and g are said to be homotopic.
(2007-11-05) 1(X) The homotopy classes of the loops going through a given base point. (Same groups forarc-connected base points, up to inner isomorphism.)
(2007-11-06) Generalizing thefundamental group to n-dimensional hyperloops.
(2007-11-06) Differentiable maps with differentiable inverses.
(2007-10-31)
(J. T. of Summerville, SC.2000-11-19) How many edges (lines) are in a cylinder?
I assume we're talking about a finite cylinder; the "ordinary kind"with two parallel bases, which are usually circular (as opposed, say,to an infinite cylinder with an infinite lateral surface and no bases).
The answer is, of course, that there aretwo edges, the two circles.
I think you figured this out by yourself and did not need anybody to tell you,so I suppose yourreal concern is elsewhere...
Because you used the term "edges" I suspect you think you've found anexception to the Descartes-Euler formula, which states that "in a polyhedron"the numbers of faces (F), edges (E) and vertices (V)are related by the formula: F-E+V=2.
In a way, you have such a "counterexample": In a cylinder, there are 3 faces(top, bottom, lateral), 2 edges (top and bottom circles) and no vertices,so that F-E+V is 1, not 2! What could be wrong?
Nothing is wrong if things are precisely stated.Edges and faces are allowed to be curved, but the Descartes-Euler formulahas 3 restrictions, namely:
It only applies to a (polyhedral) surface which is topologically "like" a sphere(imagine making the polyhedron out of flexible plastic and blowing air into it,and you'll see what I mean). Your cylinderdoes qualify (a torus would not).
It only applies if all faces are "like" an open disk.The top and bottom faces of your cylinder do qualify, but the lateral facedoes not.
It only applies if all edges are "like" an open line segment.Neither of your circular edges qualifies.
There are two ways to fix the situation.The first one is to introduce new edges and verticesartificially tomeet the above 3 conditions. For example, put a new vertex on the top edge and on the bottom edge.This satisfies condition (3),since a circle minus a point is "like" an open line segment. The remaining problem is condition (2); the lateral face is not "like"an open disk (or square, same thing). To make it so, "cut" it by introducing a regular edge betweenyour two new vertices. Now that all 3 conditions are met, what do we have?3 faces, 3 edges and 2 vertices. Since 3-3+2 is indeed 2, the Descartes-Euler formula does hold.
The better way to fix the formula does not involve introducingunnecessary edges or vertices. It involves the so-calledEuler characteristic,often denoted (chi):
The Euler Characteristic (chi )
The fundamental properties of (chi)may be summarized as follows:
Any set with a single element has a of 1 : x, ( {x} ) = 1
is additive: For twodisjoint sets E and F, (EF)= (E) + (F)
If E ishomeomorphic to F, then (E) = (F) ("Homeomorphic" is the precise term for topologically "like".)
Using the above 3 properties as axioms, it's easy to show by induction that, if it's defined at all, the of n-dimensionalspace can only be (-1). (: A plane divides space into 3 disjoint parts; itself and 2 others...)
(point) = 1
(entire straight line, oropen segment) = -1
(plane or open disc) = 1
(space or open ball) = -1
(space with n-dimensions) =(-1)
(surface of a sphere) = 2
(surface of an infinite cylinder) = 0
(surface of torus) = 0
(circle, orsemi-open segment) = 0
etc.
Now, back to our problem: Why is the Descartes-Euler formula valid to begin with? Well, that's because the of a sphere's surface is 2and it's "made from" disjoint faces, edges and vertices, each respectively with a of 1, -1 and 1.
In the "natural" breakdown of your cylinder (whose is indeed 2), you have no vertices, two ordinary faces (whose is 1) and one face whose is 0(the lateral face), whereas the of both edges is 0.The total count does match.
Noteto as many objects as the axioms would allow. This question does not seem to have been tackled by anyone yet... Consider, for example, the union A of all the intervals [2n,2n+1[ from an even integer (included) to the next integer (excluded). The union of two disjoint sets homeomorphic to A canbe arranged to be either the whole number line or another set homeomorphic to A. So, if (A) was defined to be x, we would have: x = x + x and 1 = x + x Thus, x cannot possibly be any ordinary number,and the latter equation says x is nothing like a signed infinity either [as1].At best, x could be defined as an unsigned infinity() like the"infinite circle" at the horizon of the complex plane( is undetermined). This could be a hint that a proper extension of wouldhave complex values...
(2003-11-27) Extending the Euler characteristic (1752) to complex values.
Just about 3 years after posting the previous articleat itsoriginal location, we resumed our reflectionabout an extended Euler characteristic. The hunch about complex values turned out to be decisive,based on our previous observation that the of the set Adescribed in the footnote could only be anunsigned infinity...
The set A was clearly a failed attempt at building something with a of ½. [As I recall, finding out it could only be an unsigned infinity wasdisappointing...] With hindsight, it's clear that there's a more compelling approach, based on anotherwell-known property of concerning cartesian products,which is worth preserving in any interesting extension of :
( E F ) = (E) (F)
Using the 3 "axioms" of the previous article [and the value(-1)n which they impose for the of ordinaryn-dimensional Euclidean space]this relation can be easily established by [structural] inductionfor all "polyhedral" sets. (Such sets, which are theusualdomain of definition of ,consist of finite unions ofdisjoint components,each homeomorphic to some n-dimensional Euclidean space,which are called its vertices, edges, faces, cells...) Therefore, the above relationdoes not contradict our three axioms and may beuse as afourth axiom in a larger scope of more general sets,which remains to be defined...
As we expect complex numbers to be involved,we're also expecting an arbitrary choice betweeni and-i,probably linked to the chirality of sets so thatthe chi of a set and of its "mirror image" are complexconjugates of each other. We are thus led to assume that is only preservedby homeomorphisms that conserve chirality and could restate the third axiom (C) accordingly,in terms of those homeomorphism which preserve theorientation of an immersing space.
For an homeomorphism which does not preserve such an orientation, it may be possible tofind a larger space in which the orientation is preserved whose restriction to a smallerspace violates orientation (a two-dimensional symmetry about a line isa restriction to the plane of a three-dimensional rotation about that line). This is a clue that an intrinsically chiral topological spacecan't be immersed in a space of finitely many "dimensions".
Let's try to build a set E whose cartesian squareEE has a of -1... We would then expect the cartesian product of E and its mirror image to have a of +1 and this may guide the search...
Consider aHilbert space with the countable basisdenoted |0>, |1>, |2>, |3>, etc. It is homeomorphic to its own cartesian square (: Use the even coordinates of a given ket to forma first ket and the odd ones to form a second ket.)
(2007-10-31) Werner Boy found a 3D immersion of the real projective plane.
Theset whose elements are straight lines going throughthe origin in three-dimensional Euclidean space is knownas the real projective plane.
David Hilbert (1862-1943)did not think the real projective plane could be immersed as an ordinary surface in 3-dimensional Euclidean space,but he couldn't prove it was impossible. So, he assigned the task to one of his graduate students, Werner Boy, who earned his Ph.D. in1901 by finding such an immersion, now called Boy's surface.
Boy's surface hasEuler characteristic = 1. It can be represented as a single-sided surface with a vertical axis of ternary symmetry. On that axis is a single pole P which looks like an ordinary point from the topbut appears from the bottom as thetriple point T where three seams meet (each such seam is locally equivalent to three flat surfacessharing an edge, two of those can be smoothly aligned to allow a vantage pointwhere all seams are hidden behind a smooth part of the surface). Other representations do not break at all any fundamental ternary symmetry.
(2017-07-25) A closed surface is a compact 2-D manifold without boundary.
A (topological) surface is simply a two-dimensional manifold. Unfortunately, the traditional locution closed surface doesn'tnecessarily apply to a surface which is merelyclosed;it has to becompact and borderless as well (i.e., itsboundary must be empty).
Connected Sum of two Surfaces :
The connected-sum of two surfaces is obtained by removing an open disk from oneand a closed disk from the other and gluing the two remaining component so that the two deleted disk share the same border.
(2006-08-26) About the additive continuous functions of d-dimensional rigid bodies.
All the finite unions ofconvex sets of pointsin d-dimensional Euclidean space form what's called the d-dimensional convex ring (a set of points isconvex if it contains all thestraight segments whose extremities are in it).
A map is said to be conditionally continuous when the imagesof the convex approximations to a convex body always converges to the image of the body. The blog ofDan Piponi may serve as aniceintroduction to Hadwiger's Theorem, a celebrated theorem of integral geometry published in 1957 by the Swissmathematician Hugo Hadwiger (1908-1981). Namely:
If it is invariant under translations and rotations in d-dimensional euclidean space,any finitely additive function which maps finite unions of convex bodiesto real numbers must be a linear combination of d+1 uniquely defined "n-dimensional content" functions (where the index n goes from 0 to d).
Hadwiger's "0-dimensional content" function is proportional to the Euler characteristic (the chi-function ) discussedabove. In 3 dimensions, the "n-dimensional contents" for n=1,2,3 are respectivelyproportional to the body's mean curvature, its surface or its volume.
(2006-04-24) A topological definition of orientation applies to some spaces.
(2007-11-04) Counterclockwise turns of a planar curve around an outside point.
Consider a continuous oriented planar curvewhich doesn't go through point O.
: [a,b ] 2-{O} = *
As the point M = (t) moves positively along that curve (which neednot be differentiable) an observer at the origin O mayrecord unambiguously the variations of the angle which OM forms withsome fixed direction The usual ambiguity modulo2 does not apply because we are consideringa continuous variation in an angular difference which starts at zero.
The total change in that angle, expressed in turns (the number of radians divided by 2) is called the winding number W ( , O ) of around point O.
If the curve is closed, that winding number is an integer. For example, it's +1 for any counterclockwise circle going aroundthe origin. It's -1 if such a circle is oriented clockwise. It's 0 if the circle does not go around the origin.
The winding number around the origin is invariant under reparametrization of the curveand also underhomotopy within the punctured plane *. This is illustrated by the following popular theorem:
If a man and a dog walk respectively around closed curves 0 and 1 so thatthe "leash" segment [ 0(t), 1(t) ] never touches the "hydrant" O, then:
W ( 0, O ) = W ( 1, O )
This is a consequence of the invariance of the winding numberby homotopy, since the following curve is a validhomotopic interpolationwithin the punctured plane (since (t) is never on the"hydrant", because it's a point of the "leash").
(t,s) = (1-s)0(t) + s1(t)
In particular, the lemma applies whenever the distance from the hydrantto the man (or to the dog) is greater than the lengthof the leash (such an inequality ensures that the hydrantcannot be between the man and the dog).
Any complexpolynomial P of degree n > 0 has at least onecomplex root.
Proof :WLG, assume that P is a monic polynomial of leading term xn. We aim to apply the abovedog on a leash lemma.
For some positive number r, let's consider the following closedcurves (as the parameter t goes from 0 to 2).
0(t) = ( re it) n 1(t) = P ( re it)
The winding number of 0 around the origin is clearly n because the argumentof 0(t) increases by 2n when t increases continuouslyby 2.
Now, |1(t)-0(t)| is bounded by a fixed polynomial in r of degree n-1. For a large enough r, that's less than|0(t)| = r. Therefore, by thedog-leash lemma, the winding number of 1 around the origin is n (the same as 0 ).
If P didn't have any zeroes, then (t,s) = P ( r (1-s)e it )would be a valid homotopic interpolation (within the punctured complex plane) shrinking 1 down to a pointlike curve located at P(0).This would make the winding number of 1 equal to 0 instead of n. Therefore, P must have at least one zero.
(2007-11-11) For some sets, all continuous mappings have a fixed point.
A fixed-point of a mapping f isa point x such that f (x) = x. An idiomatic way to state the same thing is to say that f fixes x.
The archetypal fixed-point theorem is Brouwer's fixed-point theorem, which says that any continuousfunction from E to itself must have a fixed point when E ishomeomorphicto a closed ball of an n-dimensional Euclidean space.
Compact convex (Schauder)
Locally convex (Tychonoff)
In a generaltopological space,it may be convenient to turn the fixed-point theorem into a definition of a specific class of sets. Let's just call Brouwerian a set E for which all continuousfunctions from E to itself have at least one fixed point. The above fixed-point theorems can be expressed by stating that the following sets are Brouwerian :
A set homeomorphic to an n-dimensional closed ball. (Brouwer)
A convex compact subset of a normed vector space. (Schauder)
A locally convex subset of a topological vector space. (Tychonoff)
(2007-11-04) Thewinding number ofthe nonvanishing tangent to a planar curve.
If a point moves in the plane with a continuous nonvanishing velocity,thewinding number of the velocityabout zero-velocity is well-defined (it would not be for a curve with singular points where thevelocity can vanish, because the winding number is not definedfor a curve which goes through the origin).
This number does not depends on the details of the motion, exceptits orientation. It's a characteristic of the orientedtrajectory called the turning number. For a closed trajectory, that turning number is an integer.
One way to compute this integer for a closed curveis to focus only on those points wherethe oriented tangent has a given direction (e.g., due east). Each such point is assigned a zero value if it's an inflection point,a value +1 if the curve lays to the left of its tangent ora value of -1 if it lays to its right. The sum of those values is equal to the curve's turning number.
The Whitney-Graustein theorem states that two closeddifferentiable curves are homotopic within the plane if and only if they have the same turning number.
(2007-10-26) A regular homotopy can turn a sphere inside out.
The approach presented in the previous article for closed planar curvescan be adapted to three-dimensional orientedsmooth surfaces by focusingonly on those points where the normal vector is verticaland pointing upward. Such points are counted for +1 if the surface does not crossits tangent plane, whereas proper saddlepoints are counted as -1. The sum of all such values is a characteristic of the surface whichremains invariant by any homotopy.
In 1957,Steve Smale (b. 1930, Fields Medal in 1966) proved that an eversion of the sphere was possible (this result is so surprising that it's still known asSmale's paradox). In 1961, Arnold Shapiro came up with the first practical eversion of a sphere. He did not publish it but described it toBernard Morin. Morin discussed it with René Thom who exchangedletters about the subject with Tony Phillips. In 1966, this culminated in a popular article by Philips for Scientific American (loosely following Shapiro's original construction). In 1967, Morin came up with an eversion which was simpler than all previous ones.
In 1974, Bill Thurston(1946-2012) introduced a new sphere eversionbased on his corrugation method (illustrated in the videoOutside in).
(2007-10-31) Conway's "Zero Irrelevancy Proof" of theclassification theorem (1860).
A connected closed surface is homeomorphic to either
A sphere with n handles (orientable, = 2-2n ). Such a surface iscalled an n-torus: sphere (n=0), torus (n=1), double-torus (n=2),triple-torus, etc.
A sphere with n crosscaps (nonorientable, = 2-n ). Such a surface iscalled an n-cross surface; a term due to John Conway (1937-2020): Thereal projective plane is a cross surface,theKlein bottle is a double-cross surface,Dyck's surface is atriple-cross surface.
(2007-10-31)
There is a surjective group homomorphism from Bn to thesymmetric group Sn (the group of all permutations of n elements).Thekernel of this group is the pure braid group on n strands Pn.
* (the nonzero reals, separated into two
* (the nonzero complex numbers).
(J. T. of Summerville, SC.
