Math and stuff

Proposition Let $q: X \rightarrow Y$ be any map. For a subset $U \subset X$, show that the following are equivalent: $U$ is saturated. $U = q^{-1}(q(U))$. $U$ is a union of fibers. If $x \in U$, then every point $x’ \in X$ such that $q(x) = q(x’)$ is also...

Proposition Suppose $q: X \rightarrow Y$ is an open quotient map. Then $Y$ is Hausdorff if and only if the set $\mathscr{R} = \{ (x_1, x_2) \mid q(x_1) = q(x_2) \}$ is closed in $X \times X$. Solution First, we will suppose that $Y$ is Hausdorff. We will show that...

Proposition $I$ with the equivalence relation $0 \sim 1$ is homeomorphic to $S^1$. Solution The equivalence relation is the same as $a \sim b \iff a - b \in \mathbb{Z}$. Let $p: I \rightarrow I / \sim$ such that $p(t) = [t]$ for each $t \in I$. Let $f: I...

Proposition Let $f: X \rightarrow Y$ be a continuous bijection. If $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. Solution It suffices to show that $f$ is an open map. Let $U \subset X$ be open. We will show that $f(U)$ is open. Let $f(x) \in...

Proposition Let $X$ be any space and $Y$ be a discrete space. Show that the Cartesian product $X \times Y$ is equal to the disjoint union $\coprod_{y \in Y} X$, and the product topology is the same as the disjoint union topology. Solution \[\begin{align*} \coprod_{y \in Y} X &= \bigcup_{y...