- Source: Countably generated space
In mathematics, a topological space
X
{\displaystyle X}
is called countably generated if the topology of
X
{\displaystyle X}
is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) is determined by the convergent sequences.
The countably generated spaces are precisely the spaces having countable tightness—therefore the name countably tight is used as well.
Definition
A topological space
X
{\displaystyle X}
is called countably generated if for every subset
V
⊆
X
,
{\displaystyle V\subseteq X,}
V
{\displaystyle V}
is closed in
X
{\displaystyle X}
whenever for each countable subspace
U
{\displaystyle U}
of
X
{\displaystyle X}
the set
V
∩
U
{\displaystyle V\cap U}
is closed in
U
{\displaystyle U}
. Equivalently,
X
{\displaystyle X}
is countably generated if and only if the closure of any
A
⊆
X
{\displaystyle A\subseteq X}
equals the union of closures of all countable subsets of
A
.
{\displaystyle A.}
Countable fan tightness
A topological space
X
{\displaystyle X}
has countable fan tightness if for every point
x
∈
X
{\displaystyle x\in X}
and every sequence
A
1
,
A
2
,
…
{\displaystyle A_{1},A_{2},\ldots }
of subsets of the space
X
{\displaystyle X}
such that
x
∈
⋂
n
A
n
¯
=
A
1
¯
∩
A
2
¯
∩
⋯
,
{\displaystyle x\in {\textstyle \bigcap \limits _{n}}\,{\overline {A_{n}}}={\overline {A_{1}}}\cap {\overline {A_{2}}}\cap \cdots ,}
there are finite set
B
1
⊆
A
1
,
B
2
⊆
A
2
,
…
{\displaystyle B_{1}\subseteq A_{1},B_{2}\subseteq A_{2},\ldots }
such that
x
∈
⋃
n
B
n
¯
=
B
1
∪
B
2
∪
⋯
¯
.
{\displaystyle x\in {\overline {{\textstyle \bigcup \limits _{n}}\,B_{n}}}={\overline {B_{1}\cup B_{2}\cup \cdots }}.}
A topological space
X
{\displaystyle X}
has countable strong fan tightness if for every point
x
∈
X
{\displaystyle x\in X}
and every sequence
A
1
,
A
2
,
…
{\displaystyle A_{1},A_{2},\ldots }
of subsets of the space
X
{\displaystyle X}
such that
x
∈
⋂
n
A
n
¯
=
A
1
¯
∩
A
2
¯
∩
⋯
,
{\displaystyle x\in {\textstyle \bigcap \limits _{n}}\,{\overline {A_{n}}}={\overline {A_{1}}}\cap {\overline {A_{2}}}\cap \cdots ,}
there are points
x
1
∈
A
1
,
x
2
∈
A
2
,
…
{\displaystyle x_{1}\in A_{1},x_{2}\in A_{2},\ldots }
such that
x
∈
{
x
1
,
x
2
,
…
}
¯
.
{\displaystyle x\in {\overline {\left\{x_{1},x_{2},\ldots \right\}}}.}
Every strong Fréchet–Urysohn space has strong countable fan tightness.
Properties
A quotient of a countably generated space is again countably generated. Similarly, a topological sum of countably generated spaces is countably generated. Therefore, the countably generated spaces form a coreflective subcategory of the category of topological spaces. They are the coreflective hull of all countable spaces.
Any subspace of a countably generated space is again countably generated.
Examples
Every sequential space (in particular, every metrizable space) is countably generated.
An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of Arens–Fort space.
See also
Finitely generated space – topology in which the intersection of any family of open sets is openPages displaying wikidata descriptions as a fallback
Locally closed subset
Tightness (topology) – Function that returns cardinal numbersPages displaying short descriptions of redirect targets − Tightness is a cardinal function related to countably generated spaces and their generalizations.
References
Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78. Berlin: Springer.
External links
A Glossary of Definitions from General Topology [1]
https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf
Kata Kunci Pencarian:
- Countably generated space
- Countably generated
- First-countable space
- Compactly generated space
- Separable space
- Metrizable space
- Fréchet space
- Standard probability space
- Compact space
- Probability space
