Math and stuff

Proposition Let $G$ be a topological group. Prove that up to isomorphism, $\pi_1(G, g)$ is independent of the choice of the base point $g \in G$. Prove that $\pi_1(G, g)$ is abelian. Solution 1 Let $g_1, g_2 \in G$. Let $\phi: G \rightarrow G$ be defined such that $\phi(g) =...

Proposition Prove that a topological space $X$ is disconnected if and only if there exists a non-constant function from $X$ to the discrete space $\{ 0, 1 \}$. Solution Suppose $X$ is disconnected. Then there exist a pair of disjoint, open sets $U, V$ such that $U \cup V =...

Proposition Let $X, Y$ be topological spaces. Every constant map $f: X \rightarrow Y$ is continuous. The identity map $\text{Id}_X: X \rightarrow X$ is continuous. If $f: X \rightarrow Y$ is continuous, so is the restriction of $f$ to any open subset of $X$. Solution 1 Let $f: X \rightarrow...

Proposition A rotation of $S^1$ is a map $\rho: S^1 \rightarrow S^1$ of the form $\rho(z) = e^{i\theta}z$ for some fixed $e^{i\theta} \in S^1$. Show that if $\rho$ is a rotation, then $N(\rho \circ f) = N(f)$ for every loop $f$ in $S^1$. Solution Let $\rho(z) = e^{i\theta_0}z$ be a...

Proposition Suppose $X$ is a connected topological space, and $\sim$ is an equivalence relation on $X$ such that every equivalence class is open. Show that there is exactly one equivalence class, namely $X$ itself. Solution For each $x \in X$, let $U_x = \{ y \in X \mid x \sim...