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, apartness, and nearness, without having to define any of them in terms of the others.
A unified topological space is a set $A$ with a unified topology - three relations $\ll$, $\bowtie$, and $\approx$ between $A$ and its power set $\mathcal{P}(A)$, such that for all elements $x$ of $A$ and subsets $U$ and $V$ of $A$,
$\ll$ is a topology in the usual sense:
if $x \ll U$, then $x \in U$
if $x \ll U$ and $U \subseteq V$, then $x \ll V$
$x \ll A$
if $x \ll U$ and $x \ll V$, then $x \ll U \cap V$
if $x \ll U$, then $x \ll \{y \in A \vert y \ll U\}$
$\bowtie$ satisfies the following apartness axioms:
if $x \bowtie U$, then $x \in U$ is false
if $x \bowtie U$ and $U \subseteq V$, then $x \bowtie V$
$x \bowtie \emptyset$
if $x \bowtie U$ and $x \bowtie V$, then $x \bowtie U \cup V$
if $x \bowtie U$, then $x \bowtie \{y \in A \vert y \approx U\}$
$\approx$ satisfies the following closure space axioms:
if $x \in U$, then $x \approx U$
if $x \approx U$ and $U \subseteq V$, then $x \approx V$
$x \approx \emptyset$ is false
if $x \approx U \cup V$ and $x \approx U$, then $x \approx V$
if $x \approx \{y \in A \vert y \approx U\}$, then $x \approx U$
The following compatibility condition holds:
Unified topologies were defined in definition 10.17 of:
Created on September 6, 2024 at 11:58:07. See the history of this page for a list of all contributions to it.