The Fundamental Groups of Discrete and Indiscrete Topological Spaces

The Fundamental Groups of Discrete and Indiscrete Topological Spaces

 Theorem 1: Let $X$ be a topological space equipped with the discrete topology. Then for any $x \in X$, $\pi_1(X, x) = \{ [c_x] \}$.
• Proof: If $X$ has the discrete topology then every subset of $X$ is an open set. So the path components of $X$ are the singleton sets $\{ x \} \subseteq X$. We know that the fundamental group of a single point is the trivial group, and so:
(1)
\begin{align} \quad \pi_1(X, x) = \{ [c_x] \} \quad \blacksquare \end{align}
 Theorem 2: Let $X$] be a topological space equipped with the indiscrete topology. Then for any [[$x \in X$, $\pi_1(X, x) = \{ [c_x] \}$.
• Proof: If $X$ has the indiscrete topology then the only open subsets of $X$ are $\emptyset$ and the whole set $X$. Let $x \in X$ and let $\alpha, \beta : I \to X$ be loops such that $\alpha(0) = \alpha(1) = x$ and $\beta(0) = \beta(1) = x$. Define a function $H : X \times I \to X$ by:
(2)
\begin{align} \quad H(s, t) = \left\{\begin{matrix} \alpha(x) & \mathrm{if} \: 0 \leq s \leq 1, t = 0\\ \beta(x) & \mathrm{if} \: 0 \leq s \leq 1, t = 1 \\ x & \mathrm{else} \end{matrix}\right. \end{align}
• Then $H$ is a trivially a continuous map from $X$ having the indiscrete topology. Furthermore:
(3)
\begin{align} \quad H_0(s) = H(s, 0) = \alpha (x) \end{align}
(4)
\begin{align} \quad H_1(s) = H(s, 1) = \beta (x) \end{align}
• And:
(5)
\begin{align} \quad H_t(0) = H(0, t) = x = H(1, t) = H_t(1) \end{align}
• So $\alpha$ and $\beta$ are homotopic relative to $\{ 0, 1 \}$, so indeed, the fundamental group is:
(6)
\begin{align} \quad \pi_1(X, x) = \{ [c_x] \} \quad \blacksquare \end{align}