Math and stuff

Proposition Show that a subset $A \subset X$ is dense if and only if every nonempty open subset of $X$ contains a point of $A$. Solution Suppose $A \subset X$ is dense. Let $U \subset X$ be a nonempty open subset. Suppose that $U$ contains no point of $A$. Then...

Proposition Let $X$ be a topological space and let $A \subset X$ be any subset. A point is in $\Int{A}$ if and only if it has a neighborhood contained in $A$. A point is in $\Ext{A}$ if and only if it has a neighborhood contained in $X \setminus A$. A...

I have been listening to the audio book Make It Stick, and I decided to use some approaches presented in the book. The bean bag example The book claims that, in order to get good at a 3-foot toss, it is better to practice 2-foot and 4-foot tosses than a...

Proposition Suppose $X$ is a second countable space. Then $X$ contains a countable dense subset. Solution Let $\mathcal{B}$ be a countable basis for $X$. For each nonempty $B \in \mathcal{B}$, we will pick a point $x_B \in B$ arbitrarily. We claim that $A = \{ x_B \mid B \in \mathcal{B},...

Proposition Let $X$ be a topological space and let $\mathscr{U}$ be an open cover of $X$. Suppose we are given a basis for each $U \in \mathscr{U}$ (when considered as a topological space in its own right). Show that the union of all those bases is a basis for $X$....