Geometric Topology 1

Topological Basics

Definitions: A map $f: X \rightarrow Y$ is continuous if for any open set $U \subset Y$ this implies $f^{-1}(U)$ is open in $X$.

For $f: \mathbb{R} \rightarrow \mathbb{R}$ this coincides with the familiar notion of continuity in that topology.

A map $f: X \rightarrow Y$ is a homeomorphism if $f$ is bijective and both $f$ and $f^{-1}$ are continuous.

Beware of $f: [0,2\pi) \rightarrow S^1$ with $f(t) = (\cos(t),\sin(t))$ as $f$ is continuous but $f^{-1}$ is not.

Let $X$ be a topological space, $S$ any subset of $X$. $S$ becomes a topological space under the following topology:

$V \subset S$ is declared open if and only if $V = S \cap U$ where $U$ is open in $X$.

$S = [0,2\pi) \subset \mathbb{R} = X$ and $U = (-\pi,\pi)$ open in $\mathbb{R}$ means $V = [0,\pi)$ is open in $S$.

If $X$ is a topological space and $\sim$ is an equivalence relation on $X$, let $Q = X/\sim$ be the set of equivalence classes.

A subset $U$ of $Q$ is declared open if and only if $q^{-1}(U)$ is open in $X$, where $q: X \rightarrow Q$ maps $x$ to its equivalence class.