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
…
A $\sigma$-locale consists of a set $X$ with a sigma-frame $\mathcal{O}(X)$.
A continuous function $f:X \to Y$ between $\sigma$-locales $X$ and $Y$ is a sigma-frame homomorphism $f^*:\mathcal{O}(Y) \to \mathcal{O}(X)$, a monotonic function which preserves finite meets and countable joins.
Francesco Ciraulo, $\sigma$-locales in Formal Topology, Logical Methods in Computer Science, Volume 18, Issue 1 (January 12, 2022) (doi:10.46298/lmcs-18%281%3A7%292022, arXiv:1801.09644)
Alex Simpson, Measure, randomness and sublocales, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. (doi:10.1016/j.apal.2011.12.014)
Created on February 3, 2024 at 04:03:04. See the history of this page for a list of all contributions to it.