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Continuity/Compactness/Connectedness The Definition of Continuity In mathematical analysis, we define continuity of a function like this: $f$ is continuous at $x\_0$ if for every $\epsilon$, there exists a $\delta$ such that for every $x \in R^n$, if $\vert \vert x - x\_0 \vert \vert < \delta$, then $\vert \vert f(x) - f(x\_0) \vert \vert < \epsilon$. And we define an open set in $R^n$ as follows: $U \subset R^n$ is called an open set in $R^n$ if for every $x \in U$, there exists a $\delta$ such that $B(x;\delta) \subset U$, where $B(x;\delta)$ is an open ball. ...

The ultimate goal of topology: Classify spaces up to homeomorphism. This immediately leads to two questions: Given two spaces, how do we prove they are homeomorphic? This is often rather hard. Given two spaces, how do we prove they are not homeomorphic? This is relatively easier. The effective tools for proving that two spaces are not homeomorphic are topological invariants. Open Sets/Topologies/Continuity Open Sets Let $X$ be a set, and let $\mathscr{F}$ be a family consisting of some subsets of $X$. We call $F$ an open set in $X$ if: ...