topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
The category of cubical sets, just as with any reasonable category of shapes for higher structures, admits a geometric realisation functor to the category of topological spaces. The universal property of the category of cubes allows for a particularly economical and canonical construction of this functor compared to some other such categories of shapes.
We make use of the notation at category of cubes and cubical set, and denote the category of topological spaces by $\mathsf{Top}$. Let $I$ denote the topological unit interval, let $\bullet$ denote the topological point, let $0 : \bullet \rightarrow I$ and $1: \bullet \rightarrow I$ be the continuous map which pick out the endpoints $0$ and $1$ of $I$ respectively, and let $t : I \rightarrow \bullet$ be the canonical map.
We denote by $\left| - \right|_{\leq 1}$ the functor $\square_{\leq 1} \rightarrow \mathsf{Top}$ given by $I^{0} \mapsto \bullet$, $I^{1} \mapsto I$, $i_{0} \mapsto 0$, $i_{1} \mapsto 1$, and $p \mapsto t$.
We denote by $\left| - \right|_{\square}: \square \rightarrow \mathsf{Top}$ the canonical functor determined by $\left| - \right|_{\leq 1}$, the cartesian monoidal structure on $\mathsf{Top}$, and the universal property of $\square$.
The geometric realisation functor $\mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Top}$ is the canonical functor determined by $\left| - \right|_{\square}$, the fact that $\mathsf{Top}$ is co-complete, and the universal property of $\mathsf{Set}^{\square^{op}}$.
Last revised on August 7, 2022 at 07:17:11. See the history of this page for a list of all contributions to it.