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
In a topological space $(X,\tau)$ a point (element) $x \in X$ is called a closed point if the singleton set $\{x\} \subset X$ is a closed subset of $X$.
A topological space is a T1-space precisely if all its points are closed points.
In the Zariski topology on an algebraic variety $Spec(R)$, the closed points correspond to the maximal ideals in $R$ (this Prop.).
In particular the prime numbers correspond to the closed points in Spec(Z) (this Example).
Last revised on April 4, 2021 at 02:47:20. See the history of this page for a list of all contributions to it.