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 notion of topological space which comes with predefined notions of neighbourhood, point-set apartness, and nearness, without having to define any of them in terms of the others.
A unified topological space is a set with a unified topology - three relations , , and between and its power set , such that for all elements of and subsets and of ,
is a topology in the usual sense:
if , then
if and , then
if and , then
if , then
is a point-set apartness in the usual sense:
if , then is false
if and , then
if and , then
if , then
satisfies the following closure space axioms:
if , then
if and , then
is false
if and , then
if , then
The following compatibility condition holds:
Unified topologies were defined in definition 10.17 of:
Last revised on November 11, 2024 at 15:29:27. See the history of this page for a list of all contributions to it.