Simply connected space

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In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every 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 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 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 X is called simply connected if it is path-connected and any loop in X defined by f : S1X can be contracted to a point: there exists a continuous map F : D2X such that F restricted to S1 is f. Here, S1 and D2 denotes the unit circle and closed unit disk in the Euclidean plane respectively.

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

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

In complex analysis: an open subset $X\subseteq \mathbb {C}$ is simply connected if and only if both X 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 a nice example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. It might also be worth pointing out that a relaxation of the requirement that X be connected leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has connected extended complement exactly when each of its connected components are 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 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 but not simply connected are called non-simply connected or multiply connected. A sphere is simply connected because every loop can be contracted (on the surface) to a point.

The definition only rules out 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 any dimension, is called contractibility.

Examples 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 is also not simply connected because the purple loop cannot contract to a point without leaving the solid.

Properties

A surface (two-dimensional topological manifold) is simply connected if and only if it is connected and its genus (the number of handles of the surface) is 0.

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

If X and Y are homotopy equivalent and X is simply connected, then so is 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 C - {0}, which is not simply connected.

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

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