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 $X$, the category of open subsets $Op(X)$ of $X$ is the category whose
objects are the open subsets $U \hookrightarrow X$ of $X$;
morphisms are the inclusions
$\array{ V &&\hookrightarrow && U \\ & \searrow && \swarrow \\ && X }$ of open subsets into each other.
The category $Op(X)$ is a poset, in fact a frame (dually a locale): it is the frame of opens of $X$.
The category $Op(X)$ is naturally equipped with the structure of a site, where a collection $\{U_i \to U\}_i$ of morphisms is a cover precisely if their union in $X$ equals $U$:
The category of sheaves on $Op(X)$ equipped with this site structure is typically referred to as the category of sheaves on the topological space and denoted
The category $Op(X)$ is also a suplattice.
Last revised on March 26, 2019 at 18:09:22. See the history of this page for a list of all contributions to it.