- **Topology**: family of subsets closed under arbitrary union and finite intersection. - **Open Set**: elements of a topology. - **Closed Set**: complement of an open set. - **Interior**: largest open set contained in a set. - **Closure**: smallest closed set contained in a set. - **Boundary**: difference between interior and closure. - **Neighbourhood**: a set containing a point in its interior. - **Accumulation Point**: a point for which all of its neighbourhood intersects some set. - **Neighbourhood Base**: collection of open sets that contain a point, where for any other open set containing it, there's a set in the collection contained by that set. - **Base**: collection of open sets containing a neighbourhood base for every point. - **Dense**: a set whose closure is the whole space. - **Nowhere Dense**: a set whose interior is empty. - **First Countable**: every point has a countable neighbourhood base. - **Second Countable**: countable base. - **Separable**: countable dense subset. - **Generated Topology**: smallest topology containing a collection of sets. - **Subbase**: generating set of a topology. - **Weaker/Stronger, Coarser/Finer**: One topology contained/containing another. - $T_0$: $x \ne y$ implies that there exists an open set containing $x$ not $y$, or $y$ not $x$. - $T_1$: $x \ne y$ implies that there exists an open set containing $x$ not $y$. - $T_2$: distinct points can be separated by disjoint open sets. - $T_3$: a point and closed set can be separated by disjoint open sets. - $T_4$: disjoint closed sets can be separated by disjoint open sets. - **Continuous**: preimage of open set is open. - **Homeomorphism**: bijective homomorphism. - **Weak Topology**: given a set $X$ and a family of mappings from $X$ to some topological spaces, the weak topology on $X$ is the smallest topology on $X$ that makes all the mappings continuous. - **Product Topology:** weak topology generated by projection maps. - **Compact**: every open cover has finite subcover. - **Sequentially Compact**: every sequence has a convergent subsequence. - **Precompact**: closure is compact. - **Locally Compact:** every point has compact neighbourhood. - **Support**: closure of complement of kernel. - **Compactly Supported**: support is compact. - **Topology of Uniform Convergence on Compact Sets**: uniform norm but only measures compact sets. Use these balls to generate a topology. - **$\sigma$-compact**: countable union of compact sets.