Math and stuff

Proposition Suppose $S$ is a subspace of the topological space $X$. If $U \subset S \subset X$, $U$ is open in $S$, and $S$ is open in $X$, then $U$ is open in $X$. The same is true with “closed” in place of “open.” If $U$ is a subset of...

Proposition Any composition of quotient maps is a quotient map. An injective quotient map is a homeomorphism. If $q: X \rightarrow Y$ is a quotient map, a subset $K \subset Y$ is closed if and only if $q^{-1}(K)$ is closed in $X$. If $q: X \rightarrow Y$ is a quotient...

Proposition $\mathbb{R}^{n - 1}$ is homeomorphic to $\partial \mathbb{H}^n$. Solution Let $f: \mathbb{R}^{n - 1} \rightarrow \partial \mathbb{H}^n$ be defined such that $f(x_1, \cdots, x_{n - 1}) = (x_1, \cdots, x_{n - 1}, 0)$. Injective? $f(x) = f(y)$ implies that $(x, 0) = (y, 0)$, so $x = y$. Surjective?...

Proposition Suppose $M$ is an $n$-dimensional manifold with boundary. Show that $\partial M$ is an $(n - 1)$-manifold (without boundary) when endowed with the subspace topology. You may use without proof the fact that $\Int{M}$ and $\partial M$ are disjoint. Solution Let $M$ be an $n$-dimensional manifold with boundary. We...

Proposition A continuous surjective map $q: X \rightarrow Y$ is a quotient map if and only if it takes saturated open subsets to open subsets, or saturated closed subsets to closed subsets. Solution Suppose $q$ is a quotient map. Let $U \subset X$ be a saturated open subset. As we...