Inmathematics,general topology (orpoint set topology) is the branch oftopology that deals with the basicset-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, includingdifferential topology,geometric topology, andalgebraic topology.
The fundamental concepts in point-set topology arecontinuity,compactness, andconnectedness:
The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept ofopen sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called atopology. A set with a topology is called atopological space.
Metric spaces are an important class of topological spaces where a real, non-negative distance, also called ametric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.
General topology grew out of a number of areas, most importantly the following:
General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition ofcontinuity, in a technically adequate form that can be applied in any area of mathematics.
LetX be a set and letτ be afamily ofsubsets ofX. Thenτ is called atopology on X if:[1][2]
Ifτ is a topology onX, then the pair (X,τ) is called atopological space. The notationXτ may be used to denote a setX endowed with the particular topologyτ.
The members ofτ are calledopen sets inX. A subset ofX is said to beclosed if itscomplement is inτ (i.e., its complement is open). A subset ofX may be open, closed, both (clopen set), or neither. The empty set andX itself are always both closed and open.
Abase (orbasis)B for atopological spaceX withtopologyT is a collection ofopen sets inT such that every open set inT can be written as a union of elements ofB.[3][4] We say that the basegenerates the topologyT. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them.
Every subset of a topological space can be given thesubspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For anyindexed family of topological spaces, the product can be given theproduct topology, which is generated by the inverse images of open sets of the factors under theprojection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.
Aquotient space is defined as follows: ifX is a topological space andY is a set, and iff :X→Y is asurjectivefunction, then thequotient topology onY is the collection of subsets ofY that have openinverse images underf. In other words, the quotient topology is the finest topology onY for whichf is continuous. A common example of a quotient topology is when anequivalence relation is defined on the topological spaceX. The mapf is then the natural projection onto the set ofequivalence classes.
A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space.
Any set can be given thediscrete topology, in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given thetrivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must beHausdorff spaces where limit points are unique.
Any set can be given thecofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallestT1 topology on any infinite set.
Any set can be given thecocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.
There are many ways to define a topology onR, the set ofreal numbers. The standard topology onR is generated by theopen intervals. The set of all open intervals forms abase or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, theEuclidean spacesRn can be given a topology. In the usual topology onRn the basic open sets are the openballs. Similarly,C, the set ofcomplex numbers, andCn have a standard topology in which the basic open sets are open balls.
The real line can also be given thelower limit topology. Here, the basic open sets are the half open intervals [a,b). This topology onR is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.
Everymetric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on anynormed vector space. On a finite-dimensionalvector space this topology is the same for all norms.
Continuity is expressed in terms ofneighborhoods:f is continuous at some pointx ∈ X if and only if for any neighborhoodV off(x), there is a neighborhoodU ofx such thatf(U) ⊆ V. Intuitively, continuity means no matter how "small"V becomes, there is always aU containingx that maps insideV and whose image underf containsf(x). This is equivalent to the condition that thepreimages of the open (closed) sets inY are open (closed) inX. In metric spaces, this definition is equivalent to theε–δ-definition that is often used in analysis.
An extreme example: if a setX is given thediscrete topology, all functions
to any topological spaceT are continuous. On the other hand, ifX is equipped with theindiscrete topology and the spaceT set is at leastT0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.
Severalequivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.
Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms ofneighborhoods:f is continuous at some pointx ∈ X if and only if for any neighborhoodV off(x), there is a neighborhoodU ofx such thatf(U) ⊆ V. Intuitively, continuity means no matter how "small"V becomes, there is always aU containingx that maps insideV.
IfX andY are metric spaces, it is equivalent to consider theneighborhood system ofopen balls centered atx andf(x) instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance.
Note, however, that if the target space isHausdorff, it is still true thatf is continuous ata if and only if the limit off asx approachesa isf(a). At anisolated point, every function is continuous.
In several contexts, the topology of a space is conveniently specified in terms oflimit points. In many instances, this is accomplished by specifying when a point is thelimit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by adirected set, known asnets.[5] A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.
In detail, a functionf:X →Y issequentially continuous if whenever a sequence (xn) inX converges to a limitx, the sequence (f(xn)) converges tof(x).[6] Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. IfX is afirst-countable space andcountable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, ifX is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are calledsequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.
Instead of specifying the open subsets of a topological space, the topology can also be determined by aclosure operator (denoted cl), which assigns to any subsetA ⊆X itsclosure, or aninterior operator (denoted int), which assigns to any subsetA ofX itsinterior. In these terms, a function
between topological spaces is continuous in the sense above if and only if for all subsetsA ofX
That is to say, given any elementx ofX that is in the closure of any subsetA,f(x) belongs to the closure off(A). This is equivalent to the requirement that for all subsetsA' ofX'
Moreover,
is continuous if and only if
for any subsetA ofX.
Iff:X →Y andg:Y →Z are continuous, then so is the compositiong ∘f:X →Z. Iff:X →Y is continuous and
The possible topologies on a fixed setX arepartially ordered: a topology τ1 is said to becoarser than another topology τ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, theidentity map
is continuous if and only if τ1 ⊆ τ2 (see alsocomparison of topologies). More generally, a continuous function
stays continuous if the topology τY is replaced by acoarser topology and/or τX is replaced by afiner topology.
Symmetric to the concept of a continuous map is anopen map, for whichimages of open sets are open. In fact, if an open mapf has aninverse function, that inverse is continuous, and if a continuous mapg has an inverse, that inverse is open. Given abijective functionf between two topological spaces, the inverse functionf−1 need not be continuous. A bijective continuous function with continuous inverse function is called ahomeomorphism.
If a continuous bijection has as itsdomain acompact space and itscodomain isHausdorff, then it is a homeomorphism.
Given a function
whereX is a topological space andS is a set (without a specified topology), thefinal topology onS is defined by letting the open sets ofS be those subsetsA ofS for whichf−1(A) is open inX. IfS has an existing topology,f is continuous with respect to this topology if and only if the existing topology iscoarser than the final topology onS. Thus the final topology can be characterized as the finest topology onS that makesf continuous. Iff issurjective, this topology is canonically identified with thequotient topology under theequivalence relation defined byf.
Dually, for a functionf from a setS to a topological spaceX, theinitial topology onS has a basis of open sets given by those sets of the formf−1(U) whereU is open inX . IfS has an existing topology,f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology onS. Thus the initial topology can be characterized as the coarsest topology onS that makesf continuous. Iff is injective, this topology is canonically identified with thesubspace topology ofS, viewed as a subset ofX.
A topology on a setS is uniquely determined by the class of all continuous functions into all topological spacesX.Dually, a similar idea can be applied to maps
Formally, atopological spaceX is calledcompact if each of itsopen covers has afinitesubcover. Otherwise it is callednon-compact. Explicitly, this means that for every arbitrary collection
of open subsets ofX such that
there is a finite subsetJ ofA such that
Some branches of mathematics such asalgebraic geometry, typically influenced by the French school ofBourbaki, use the termquasi-compact for the general notion, and reserve the termcompact for topological spaces that are bothHausdorff andquasi-compact. A compact set is sometimes referred to as acompactum, pluralcompacta.
Every closedinterval inR of finite length iscompact. More is true: InRn, a set is compactif and only if it isclosed and bounded. (SeeHeine–Borel theorem).
Every continuous image of a compact space is compact.
A compact subset of a Hausdorff space is closed.
Every continuousbijection from a compact space to a Hausdorff space is necessarily ahomeomorphism.
Everysequence of points in a compact metric space has a convergent subsequence.
Every compact finite-dimensionalmanifold can be embedded in some Euclidean spaceRn.
Atopological spaceX is said to bedisconnected if it is theunion of twodisjointnonemptyopen sets. Otherwise,X is said to beconnected. Asubset of a topological space is said to be connected if it is connected under itssubspace topology. Some authors exclude theempty set (with its unique topology) as a connected space, but this article does not follow that practice.
For a topological spaceX the following conditions are equivalent:
Every interval inR isconnected.
The continuous image of aconnected space is connected.
Themaximal connected subsets (ordered byinclusion) of a nonempty topological space are called theconnected components of the space.The components of any topological spaceX form apartition of X: they aredisjoint, nonempty, and their union is the whole space.Every component is aclosed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of therational numbers are the one-point sets, which are not open.
Let be the connected component ofx in a topological spaceX, and be the intersection of all open-closed sets containingx (calledquasi-component ofx.) Then where the equality holds ifX is compact Hausdorff or locally connected.
A space in which all components are one-point sets is calledtotally disconnected. Related to this property, a spaceX is calledtotally separated if, for any two distinct elementsx andy ofX, there exist disjointopen neighborhoodsU ofx andV ofy such thatX is the union ofU andV. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbersQ, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not evenHausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.
Apath from a pointx to a pointy in atopological spaceX is acontinuous functionf from theunit interval [0,1] toX withf(0) =x andf(1) =y. Apath-component ofX is anequivalence class ofX under theequivalence relation, which makesx equivalent toy if there is a path fromx toy. The spaceX is said to bepath-connected (orpathwise connected or0-connected) if there is at most one path-component; that is, if there is a path joining any two points inX. Again, many authors exclude the empty space.
Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extendedlong lineL* and thetopologist's sine curve.
However, subsets of thereal lineR are connectedif and only if they are path-connected; these subsets are theintervals ofR. Also,open subsets ofRn orCn are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same forfinite topological spaces.
GivenX such that
is the Cartesian product of the topological spacesXi,indexed by, and thecanonical projectionspi :X →Xi, theproduct topology onX is defined as thecoarsest topology (i.e. the topology with the fewest open sets) for which all the projectionspi arecontinuous. The product topology is sometimes called theTychonoff topology.
The open sets in the product topology are unions (finite or infinite) of sets of the form, where eachUi is open inXi andUi ≠ Xi only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of theXi gives a basis for the product.
The product topology onX is the topology generated by sets of the formpi−1(U), wherei is inI andU is an open subset ofXi. In other words, the sets {pi−1(U)} form asubbase for the topology onX. Asubset ofX is open if and only if it is a (possibly infinite)union ofintersections of finitely many sets of the formpi−1(U). Thepi−1(U) are sometimes calledopen cylinders, and their intersections arecylinder sets.
In general, the product of the topologies of eachXi forms a basis for what is called thebox topology onX. In general, the box topology isfiner than the product topology, but for finite products they coincide.
Related to compactness isTychonoff's theorem: the (arbitrary)product of compact spaces is compact.
Many of these names have alternative meanings in some of mathematical literature, as explained onHistory of the separation axioms; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous.
Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles.
In all of the following definitions,X is again atopological space.
TheTietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
Anaxiom of countability is aproperty of certainmathematical objects (usually in acategory) that requires the existence of acountable set with certain properties, while without it such sets might not exist.
Important countability axioms fortopological spaces:
Relations:
Ametric space[7] is anordered pair where is a set and is ametric on, i.e., afunction
such that for any, the following holds:
The function is also calleddistance function or simplydistance. Often, is omitted and one just writes for a metric space if it is clear from the context what metric is used.
Everymetric space isparacompact andHausdorff, and thusnormal.
Themetrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
TheBaire category theorem says: IfX is acomplete metric space or alocally compact Hausdorff space, then the interior of every union ofcountably manynowhere dense sets is empty.[8]
Any open subspace of aBaire space is itself a Baire space.
Acontinuum (plcontinua) is a nonemptycompactconnectedmetric space, or less frequently, acompactconnectedHausdorff space.Continuum theory is the branch of topology devoted to the study of continua. These objects arise frequently in nearly all areas of topology andanalysis, and their properties are strong enough to yield many 'geometric' features.
Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change. Many examples with applications to physics and other areas of math includefluid dynamics,billiards andflows on manifolds. The topological characteristics offractals in fractal geometry, ofJulia sets and theMandelbrot set arising incomplex dynamics, and ofattractors in differential equations are often critical to understanding these systems.[citation needed]
Pointless topology (also calledpoint-free orpointfree topology) is an approach totopology that avoids mentioning points. The name 'pointless topology' is due toJohn von Neumann.[9] The ideas of pointless topology are closely related tomereotopologies, in which regions (sets) are treated as foundational without explicit reference to underlying point sets.
Dimension theory is a branch of general topology dealing withdimensional invariants oftopological spaces.
Atopological algebraA over atopological fieldK is atopological vector space together with a continuous multiplication
that makes it analgebra overK. A unitalassociative topological algebra is atopological ring.
The term was coined byDavid van Dantzig; it appears in the title of hisdoctoral dissertation (1931).
Intopology and related areas ofmathematics, ametrizable space is atopological space that ishomeomorphic to ametric space. That is, a topological space is said to be metrizable if there is a metric
such that the topology induced byd is.Metrization theorems aretheorems that givesufficient conditions for a topological space to be metrizable.
Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent ofZermelo–Fraenkel set theory (ZFC). A famous problem isthe normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.
Definition. A collectionB of subsets of a topological space(X,T) is called abasis forT if every open set can be expressed as a union of members ofB.
Suppose we have a topology on a setX, and a collection of open sets such that every open set is a union of members of. Then is called abase for the topology...
Some standard books on general topology include:
ThearXiv subject code ismath.GN.
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