Math and stuff

Proposition Let $U \subset \mathbb{R}^n$ be an open subset. Let $f: U \rightarrow \mathbb{R}^k$ be any continuous map. Let $f_1, \cdots, f_k$ denote each coordinate function of $f$. Let $F: U \rightarrow \mathbb{R}^{n + k}$ denote the mapping $(x_1, \cdots, x_n) \mapsto (x_1, \cdots, x_n, f_1(x), \cdots, f_k(x))$. Let $M...

Proposition Show that a subset of a topological space is closed if and only if it contains all of its limit points. Solution Suppose $A$ is closed. Let $x \notin A$. $A^c$ is a neighborhood of $x$ that does not contain a point of $A$. Thus $x$ is not a...

Proposition Suppose $S$ is a subspace of $X$. Prove that a subset $B \subset S$ is closed in $S$ if and only if it is equal to the intersection of $S$ with some closed subset of $X$. Solution Let $B \subset S$. Suppose that $B$ is closed in $S$. Then...

Proposition Suppose $X$ is a first-countable space, $A$ is any subset of $X$, and $x$ is any point of $X$. $x \in \overline{A}$ if and only if $x$ is a limit point of a sequence of points in $A$. $x \in \Int{A}$ if and only if every sequence in $X$...

Proposition Let $X$ be a second countable topological space. Show that every collection of disjoint open subsets of $X$ is countable. Solution Let $B = \{ B_1, B_2, \cdots \}$ be a countable basis of $X$. Without loss of generality, assume each $B_i$ is nonempty. It suffices to show that...