Countably generated space explained

X is called countably generated if the topology of 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 is used as well.

Definition

A topological space

X is called if for every subset V \subseteq X, V is closed in X whenever for each countable subspace U of X the set V \cap U is closed in U . Equivalently, X is countably generated if and only if the closure of any A \subseteq X equals the union of closures of all countable subsets of A.

Countable fan tightness

A topological space

X has if for every point x \in X and every sequence A _{1,} A _{2,} \ldots of subsets of the space X such that x \in { style c ap\limits _{n}} \overline{A _{n}} = \overline{A _{1}} \cap \overline{A _{2}} \cap … , there are finite set B _{1\subseteq} A _{1,} B _{2} \subseteq A _{2,} \ldots such that x \in \overline{{ style c up\limits _{n}} B _{n}} = \overline{B _{1} \cup B _{2} \cup … }.

A topological space

X has if for every point x \in X and every sequence A _{1,} A _{2,} \ldots of subsets of the space X such that x \in { style c ap\limits _{n}} \overline{A _{n}} = \overline{A _{1}} \cap \overline{A _{2}} \cap … , there are points x _{1} \in A _{1,} x _{2} \in A _{2,} \ldots such that 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

• • • − Tightness is a cardinal function related to countably generated spaces and their generalizations.

References

• Book: Herrlich, Horst. Horst Herrlich

. Horst Herrlich. Topologische Reflexionen und Coreflexionen. Springer. Berlin. 1968. Lecture Notes in Math. 78.

External links

• A Glossary of Definitions from General Topology https://web.archive.org/web/20120204044336/https://math.berkeley.edu/~apollo/topodefs.ps
• https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf