Math and stuff

Proposition Show that “homeomorphic” is an equivalence relation on the class of all topological spaces. Solution For any topological space $X$, $\Id_X$ is a homeomorphism from $X$ to $X$. Let $\phi: X \rightarrow Y$ be a homeomorphism. Then $\phi^{-1}$ is a continuous bijection. Moreover $(\phi^{-1})^{-1} = \phi$, so $\phi^{-1}$ has...

Proposition Suppose $X$ is a topological space, $A$ is a subset of $X$, and $(x_i)$ is a sequence of points in $A$ that converges to a point $x \in X$. Show that $x \in \overline{A}$. Solution Suppose $x \notin \overline{A}$. Then $\overline{A}^c$ is a neighborhood of $x$. Since $x_i \rightarrow...

Proposition Let $X, Y$, and $Z$ be topological spaces. Every constant map $f: X \rightarrow Y$ is continuous. The identity map $\Id_X: X \rightarrow X$ is continuous. If $f: X \rightarrow Y$ is continuous, so is the restriction of $f$ to any open subset of $X$. Solution 1 Let $V...

Proposition Let $X$ be a discrete topological space. Show that the only convergent sequences in $X$ are the ones that are eventually constant, that is, sequences $(x_i)$ such that $x_i = x$ for all but finitely many $i$. Solution It is easy to see that eventually constant sequences converge. Let...

Getting a PhD is hard, but applying to a PhD program is also very hard! 1: Find 3 professors for recommendation letters! This will be by far the most important aspect of your application. Make sure: You have 3 professors! Almost every school in the US requires 3 letters. I...