Simply Connected Topological Spaces

Simply Connected Topological Spaces

Recall from The Fundamental Group of a Topological Space at a Point page that if $X$ is a topological space and $x \in X$ then the fundamental group of $X$ at $x$ is:

\begin{align} \quad \pi_1(X, x) = \{ [\alpha] : \alpha(0) = x = \alpha(1) \} \end{align}

With the group operation of homotopy class multiplication defined for all $\alpha, \beta \in \pi_1(X, x)$ by:

\begin{align} \quad [\alpha][\beta] = [\alpha\beta] \end{align}

Sometimes the fundamental group of a topological space $X$ at a point $x$ is the trivial group consisting only of the homotopy class of the constant loop, $[c_x]$. Such spaces for which the fundamental groups all trivial are given a special name.

Definition: Let $X$ be a topological space. Then $X$ is said to be Simply Connected if $\pi_1(X, x) = \{ [c_x] \}$ for every $x \in X$.

The following theorem give us some examples of simply connected spaces.

Theorem 1: If $X$ is a convex topological subspace of $\mathbb{R}^n$ then $X$ is simply connected.

For example, consider the topological subspace of the closed unit disk $D^2 = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1 \}$. Clearly $D^2$ is convex and so for every $\vec{x} \in D^2$:

\begin{align} \quad \pi_1(X, \vec{x}) = \{ [c_{\vec{x}}] \} \end{align}

So $D^2$ is simply connected.

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