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
Given a topological space $X$, then a set of subset $\{S_i \subset X\}_{i \in I}$ is locally finite if every point $x \in X$ intersects only a finite number of the $S_i$.
Often this property is considered for open covers, see at locally finite open cover. But the condition also plays a role for collections of subsets which are not open or not covering, for instance in Michael's theorem (Michael 53, theorem 1).
Last revised on March 21, 2021 at 04:13:34. See the history of this page for a list of all contributions to it.