Editing Simply connected space (section)

==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]] <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]] and closed [[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]] <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]] of <math>X</math> at each point is trivial, i.e. consists only of the [[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]]s <math>\operatorname{Hom}_{\Pi(X)}(x,y)</math> in the [[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]]: 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.