View source for Simply connected space

{{short description|Space which has no holes through it}}
In [[Topology|topology]], a [[Topological space|topological space]] is called '''simply connected''' (or '''1-connected''', or '''1-simply connected'''<ref>{{Cite web|url=https://ncatlab.org/nlab/show/n-connected+space|title=n-connected space in nLab|website=ncatlab.org|access-date=2017-09-17}}</ref>) if it is path-connected and every [[Path (topology)|path]] between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The [[Fundamental group|fundamental 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==
[[Image:Runge theorem.svg|thumb|This shape represents a set that is not simply connected, because any loop that encloses one or more of the holes cannot be contracted to a point without exiting the region.]]
A [[Topological space|topological space]] <math>X</math> is called {{em|simply connected}} if it is path-connected and any [[Loop (topology)|loop]] in <math>X</math> defined by <math>f : S^1 \to X</math> can be contracted to a point: there exists a continuous map <math>F : D^2 \to X</math> such that <math>F</math> restricted to <math>S^1</math> is <math>f.</math> Here, <math>S^1</math> and <math>D^2</math> denotes the [[Unit circle|unit circle]] and closed [[Unit disk|unit disk]] in the [[Euclidean space|Euclidean plane]] respectively.

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

A topological space <math>X</math> is simply connected if and only if <math>X</math> is path-connected and the [[Fundamental group|fundamental group]] of <math>X</math> at each point is trivial, i.e. consists only of the [[Identity element|identity element]]. Similarly, <math>X</math> is simply connected if and only if for all points <math>x, y \in X,</math> the set of [[Morphism|morphism]]s <math>\operatorname{Hom}_{\Pi(X)}(x,y)</math> in the [[Fundamental groupoid|fundamental groupoid]] of <math>X</math> has only one element.<ref>{{Cite book|title=Topology and Groupoids.| last=Ronald|first=Brown| date=June 2006|publisher=CreateSpace| others=Academic Search Complete.| isbn=1419627228|location=North Charleston | oclc=712629429}}</ref>

In [[Complex analysis|complex analysis]]: an open subset <math>X \subseteq \Complex</math> is simply connected if and only if both <math>X</math> and its complement in the [[Riemann 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 <math>X</math> 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==

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 are [[Connected space|connected]] but not simply connected are called '''non-simply connected''' or '''multiply connected'''.

[[Image:P1S2all.jpg|thumb|center|400px|A [[Sphere|sphere]] is simply connected because every loop can be contracted (on the surface) to a point.]]

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To illustrate the notion of simple connectedness, suppose we are considering an object in three dimensions; for example, an object in the shape of a box, a doughnut, or a corkscrew. Think of the object as a strangely shaped aquarium full of water, with rigid sides. Now think of a diver who takes a long piece of string and trails it through the water inside the aquarium, in whatever way he pleases, and then joins the two ends of the string to form a closed loop. Now the loop begins to contract on itself, getting smaller and smaller. (Assume that the loop magically knows the best way to contract, and won't get snagged on jagged edges if it can possibly avoid them.) If the loop can always shrink all the way to a point, then the aquarium's interior {{em|is}} simply connected. If sometimes the loop gets caught&mdash;for example, around the central hole in the doughnut&mdash;then the object is {{em|not}} simply connected.
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The definition rules out only [[Handle decomposition|handle]]-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 of {{em|any}} dimension, is called [[Contractible space|contractibility]].
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Removed pending making-sense-ness. See Talk.
==Naming==
* In [[Topology|topology]], a connection is often referred to as a {{em|handle}}. This is probably a reference to the way that a (singly connected) beaker can be topologically turned into a (doubly connected) teacup by the addition of a handle.
* In [[Physics:Theoretical physics|theoretical physics]], an additional connection is known as a ''[[Physics:Wormhole|wormhole]]''.
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==Examples==
[[Image:Torus cycles.png|thumb|right|150px|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. A [[Solid torus|solid torus]] is also not simply connected because the purple loop cannot contract to a point without leaving the solid.]]
* The [[Euclidean space|Euclidean plane]] <math>\R^2</math> is simply connected, but <math>\R^2</math> minus the origin <math>(0, 0)</math> is not. If <math>n > 2,</math> then both <math>\R^n</math> and <math>\R^n</math> minus the origin are simply connected.
* Analogously: the [[N-sphere|''n''-dimensional sphere]] <math>S^n</math> is simply connected if and only if <math>n \geq 2.</math>
* Every convex subset of <math>\R^n</math> is simply connected.
* A [[Torus|torus]], the (elliptic) cylinder, the [[Möbius strip]], the [[Projective plane|projective plane]] and the [[Klein bottle]] are not simply connected.
* Every [[Topological vector space|topological vector space]] is simply connected; this includes [[Banach space]]s and [[Hilbert space]]s.
* For <math>n \geq 2,</math> the special orthogonal group <math>\operatorname{SO}(n, \R)</math> is not simply connected and the [[Special unitary group|special unitary group]] <math>\operatorname{SU}(n)</math> is simply connected.
* The one-point compactification of <math>\R</math> is not simply connected (even though <math>\R</math> is simply connected).
* The [[Long line (topology)|long line]] <math>L</math> is simply connected, but its compactification, the extended long line <math>L^*</math> is not (since it is not even path connected).

==Properties==
A surface (two-dimensional topological [[Manifold|manifold]]) is simply connected if and only if it is connected and its [[Genus (mathematics)|genus]] (the number of {{em|handles}} of the surface) is 0.

A universal cover of any (suitable) space <math>X</math> is a simply connected space which maps to <math>X</math> via a covering map.

If <math>X</math> and <math>Y</math> are homotopy equivalent and <math>X</math> is simply connected, then so is <math>Y.</math>

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 <math>\Complex \setminus \{ 0 \},</math> which is not simply connected.

The notion of simple connectedness is important in [[Complex analysis|complex analysis]] because of the following facts: 
* The [[Cauchy's integral theorem]] states that if <math>U</math> is a simply connected open subset of the [[Complex number|complex plane]] <math>\Complex,</math> and <math>f : U \to \Complex</math> is a [[Holomorphic function|holomorphic function]], then <math>f</math> has an [[Antiderivative (complex analysis)|antiderivative]] <math>F</math> on <math>U,</math> and the value of every [[Line integral|line integral]] in <math>U</math> with integrand <math>f</math> depends only on the end points <math>u</math> and <math>v</math> of the path, and can be computed as <math>F(v) - F(u).</math> The integral thus does not depend on the particular path connecting <math>u</math> and <math>v,</math> 
* The [[Riemann mapping theorem]] states that any non-empty open simply connected subset of <math>\Complex</math> (except for <math>\Complex</math> itself) is [[Conformal map|conformally equivalent]] to the [[Unit disk|unit disk]].

The notion of simple connectedness is also a crucial condition in the [[Poincaré conjecture]].

==See also==
*{{annotated link|Fundamental group}}
*{{annotated link|N-connected space}}
*{{annotated link|Unicoherent space}}

==References==
{{reflist}}
*{{cite book |last=Spanier |first=Edwin |title=Algebraic Topology |date=December 1994 |publisher=Springer |isbn=0-387-94426-5}}
*{{cite book |last=Conway |first=John |title=Functions of One Complex Variable I |year=1986 |publisher=Springer |isbn=0-387-90328-3}}
*{{cite book |last=Bourbaki |first=Nicolas |title=Lie Groups and Lie Algebras |year=2005 |publisher=Springer |isbn=3-540-43405-4}}
*{{cite book |last=Gamelin |first=Theodore |title=Complex Analysis |date=January 2001 |publisher=Springer |isbn=0-387-95069-9}}
*{{cite book |last=Joshi |first=Kapli |title=Introduction to General Topology |date=August 1983 |publisher=New Age Publishers |isbn=0-85226-444-5}}

[[Category:Algebraic topology]]
[[Category:Properties of topological spaces]]

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{{Sourceattribution|Simply connected space}}