Math and stuff

Proposition Suppose $X$ is a topological space and $A \subset B \subset X$. Show that $A$ is dense in $X$ if and only if $A$ is dense in $B$ and $B$ is dense in $X$. Solution Suppose that $A$ is dense in $X$. Since the closure of a set is...

Proposition Suppose $X$ is a topological space and $U \subset S \subset X$. Show that the closure of $U$ in $S$ is equal to $\overline{U} \cap S$. Show that the interior of $U$ in $S$ contains $\Int{U} \cap S$; give an example to show that they might not be equal....

Proposition Suppose $q: X \rightarrow Y$ is a quotient map, $Z$ is a topological space, and $f: X \rightarrow Z$ is any continuous map that is constant on the fibers of $q$ (i.e., if $q(x) = q(x’)$, then $f(x) = f(x’)$). Then there exists a unique continuous map $\tilde{f}: Y...

Proposition Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $\delta = \{ (x, x) \mid x \in X \} \subset X \times X$. Show that $X$ is Hausdorff if and only if $\delta$ is closed in $X \times X$. Solution Suppose $X$ is...

Proposition Let $X_1, \cdots, X_n$ be topological spaces. Let $\mathcal{B} = \{ U_1 \times \cdots \times U_n \mid \text{$U_i$ is an open subset of $X_i$, $i = 1, \cdots, n$} \}$. Then $\mathcal{B}$ is a basis for $X_1 \times \cdots \times X_n$. Solution Since $X_1 \times \cdots \times X_n \in...