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
typical contexts
The forgetful functor $\Gamma : Top \to Set$ from Top to Set that sends any topological space to its underlying set has a left adjoint $Disc : Set \to Top$ and a right adjoint $Codisc : Set \to Top$.
For $S \in Set$
$Disc(S)$ is the topological space on $S$ in which every subset is an open set,
this is called the discrete topology on $S$, it is the finest topology on $S$; $Disc(S)$ is called a discrete space;
$Codisc(S)$ is the topological space on $S$ whose only open sets are the empty set and $S$ itself, which is called the indiscrete topology on $S$ (rarely also antidiscrete topology or codiscrete topology or trivial topology or chaotic topology (SGA4-1, 1.1.4)), it is the coarsest topology on $S$; $Codisc(S)$ is called a indiscrete space (rarely also antidiscrete space, even more rarely codiscrete space).
For an axiomatization of this situation see codiscrete object.
Let $S$ be a set and let $(X,\tau)$ be a topological space. Then
every continuous function $(X,\tau) \longrightarrow Disc(S)$ is locally constant;
every function (of sets) $X \longrightarrow CoDisc(S)$ is continuous.
The functor $Disc$ does not preserve infinite products because the infinite product topological space of discrete spaces may be nondiscrete. Thus, $Disc$ does not have a left adjoint functor.
However, if we restrict the codomain of $Disc$ to locally connected spaces, then the left adjoint functor of $Disc$ does exist and it computes the set of connected components of a given locally connected space, i.e., is the $\pi_0$ functor.
This is discussed at locally connected spaces – cohesion over sets and cosheaf of connected components.
The terminology chaotic topology is motivated (see also at chaos) in
and via footnote 1 (page 3) in
In the context of Grothendieck topologies, this appears for instance in
following SGA4-1, 1.1.4.
Last revised on March 12, 2021 at 12:49:37. See the history of this page for a list of all contributions to it.