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
Assuming excluded middle, then:
Every Hausdorff topological space is a sober topological space.
More specifically, in a Hausdorff topological space the irreducible closed subspaces are precisely the singleton subspaces.
The second statement clearly implies the first. To see the second statement, suppose that $F$ is an irreducible closed subspace which contained two distinct points $x \neq y$. Then by the Hausdorff property there are disjoint neighbourhoods $U_x, U_y$, and hence it would follow that the relative complements $F \setminus U_x$ and $F \setminus U_y$ were distinct proper closed subsets of $F$ with
in contradiction to the assumption that $F$ is irreducible.
This proves by contradiction that every irreducible closed subset is a singleton. Conversely, generally the topological closure of every singleton is irreducible closed.
There are many examples of sober spaces which are not Hausdorff. For example, the spectrum of a ring which is not zero-dimensional is sober but not Hausdorff.
Any Hausdorff space is not only sober, but also $T_1$. However, even the converse to this fails. For example, let $X$ be the real line with a new point $p$ added, topologized such that any open set in the real line is open, and any cofinite set containing $p$ is open. Then $X$ is sober and $T_1$ but not Hausdorff.
maps from compact spaces to Hausdorff spaces are closed and proper
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
Last revised on September 20, 2018 at 19:51:10. See the history of this page for a list of all contributions to it.