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Simply connected space

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Space which has no holes through it

Intopology, atopological space is calledsimply connected (or1-connected, or1-simply connected^[1]) if it ispath-connected and everypath between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. Thefundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.

Definition and equivalent formulations

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A shape containing several holes. — This shape represents a set that isnot simply connected, because any loop that encloses one or more of the holes cannot be contracted to a point without exiting the region.

Atopological space $X {\displaystyle X}$ is calledsimply connected if it is path-connected and anyloop in $X {\displaystyle X}$ defined by $f:S^{1}\to X$ can be contracted to a point: there exists a continuous map $F:D^{2}\to X$ such that $F {\displaystyle F}$ restricted to $S^{1}$ is $f . {\displaystyle f.}$ Here, $S^{1}$ and $D^{2}$ denotes theunit circle and closedunit disk in theEuclidean plane respectively.

An equivalent formulation is this: $X {\displaystyle X}$ is simply connected if and only if it is path-connected, and whenever $p:[0,1]\to X$ and $q:[0,1]\to X$ are two paths (that is, continuous maps) with the same start and endpoint ( $p(0)=q(0)$ and $p(1)=q(1)$ ), then $p {\displaystyle p}$ can be continuously deformed into $q {\displaystyle q}$ while keeping both endpoints fixed. Explicitly, there exists ahomotopy $F:[0,1]\times [0,1]\to X$ such that $F(x,0)=p(x)$ and $F(x,1)=q(x).$

A topological space $X {\displaystyle X}$ is simply connected if and only if $X {\displaystyle X}$ is path-connected and thefundamental group of $X {\displaystyle X}$ at each point is trivial, i.e. consists only of theidentity element. Similarly, $X {\displaystyle X}$ is simply connected if and only if for all points $x,y\in X,$ the set ofmorphisms $\operatorname {Hom} _{\Pi (X)}(x,y)$ in thefundamental groupoid of $X {\displaystyle X}$ has only one element.^[2]

Incomplex analysis: an open subset $X\subseteq \mathbb {C}$ is simply connected if and only if both $X {\displaystyle X}$ and its complement in theRiemann sphere are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one furnishes an example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. A relaxation of the requirement that $X {\displaystyle X}$ be connected leads to an exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has a connected extended complement exactly when each of its connected components is simply connected.

Informal discussion

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Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with a handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that areconnected but not simply connected are callednon-simply connected ormultiply connected.

Asphere is simply connected because every loop can be contracted (on the surface) to a point.

The definition rules out onlyhandle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes ofany dimension, is calledcontractibility.

Examples

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A torus is not a simply connected surface. Neither of the two colored loops shown here can be contracted to a point without leaving the surface. Asolid torus is also not simply connected because the purple loop cannot contract to a point without leaving the solid.

TheEuclidean plane $\mathbb {R} ^{2}$ is simply connected, but $\mathbb {R} ^{2}$ minus the origin $(0,0)$ is not. If $n>2,$ then both $\mathbb {R} ^{n}$ and $\mathbb {R} ^{n}$ minus the origin are simply connected.
Analogously: then-dimensional sphere $S^{n}$ is simply connected if and only if $n\geq 2.$
Everyconvex subset of $\mathbb {R} ^{n}$ is simply connected.
Atorus, the (elliptic)cylinder, theMöbius strip, theprojective plane and theKlein bottle are not simply connected.
Everytopological vector space is simply connected; this includesBanach spaces andHilbert spaces.
For $n\geq 2,$ thespecial orthogonal group $\operatorname {SO} (n,\mathbb {R} )$ is not simply connected and thespecial unitary group $\operatorname {SU} (n)$ is simply connected.
The one-point compactification of $\mathbb {R}$ is not simply connected (even though $\mathbb {R}$ is simply connected).
Thelong line $L {\displaystyle L}$ is simply connected, but its compactification, the extended long line $L^{*}$ is not (since it is not even path connected).

Properties

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A surface (two-dimensional topologicalmanifold) is simply connected if and only if it is connected and itsgenus (the number ofhandles of the surface) is 0.

A universal cover of any (suitable) space $X {\displaystyle X}$ is a simply connected space which maps to $X {\displaystyle X}$ via acovering map.

If $X {\displaystyle X}$ and $Y {\displaystyle Y}$ arehomotopy equivalent and $X {\displaystyle X}$ is simply connected, then so is $Y . {\displaystyle Y.}$

The image of a simply connected set under a continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is $\mathbb {C} \setminus \{0\},$ which is not simply connected.

The notion of simple connectedness is important incomplex analysis because of the following facts:

Cauchy's integral theorem states that if $U {\displaystyle U}$ is a simply connected open subset of thecomplex plane $\mathbb {C} ,$ and $f:U\to \mathbb {C}$ is aholomorphic function, then $f {\displaystyle f}$ has anantiderivative $F {\displaystyle F}$ on $U, {\displaystyle U,}$ and the value of everyline integral in $U {\displaystyle U}$ with integrand $f {\displaystyle f}$ depends only on the end points $u {\displaystyle u}$ and $v {\displaystyle v}$ of the path, and can be computed as $F(v)-F(u).$ The integral thus does not depend on the particular path connecting $u {\displaystyle u}$ and $v, {\displaystyle v,}$
TheRiemann mapping theorem states that any non-empty open simply connected subset of $\mathbb {C}$ (except for $\mathbb {C}$ itself) isconformally equivalent to theunit disk.

The notion of simple connectedness is also a crucial condition in thePoincaré conjecture.

References

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^"n-connected space in nLab".ncatlab.org. Retrieved2017-09-17.
^Ronald, Brown (June 2006).Topology and Groupoids. Academic Search Complete. North Charleston: CreateSpace.ISBN 1419627228.OCLC 712629429.

Spanier, Edwin (December 1994).Algebraic Topology. Springer.ISBN 0-387-94426-5.
Conway, John (1986).Functions of One Complex Variable I. Springer.ISBN 0-387-90328-3.
Bourbaki, Nicolas (2005).Lie Groups and Lie Algebras. Springer.ISBN 3-540-43405-4.
Gamelin, Theodore (January 2001).Complex Analysis. Springer.ISBN 0-387-95069-9.
Joshi, Kapli (August 1983).Introduction to General Topology. New Age Publishers.ISBN 0-85226-444-5.

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Definition and equivalent formulations

Informal discussion

Examples

Properties

See also

References