Countably generated space

In mathematics, a topological space $$X$$ 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 $$x \in {\textstyle\bigcap\limits_n} \, \overline{A_n} = \overline{A_1} \cap \overline{A_2} \cap \cdots,$$ there are finite set $$B_1\subseteq A_1, B_2 \subseteq A_2, \ldots$$ such that $$x \in \overline{{\textstyle\bigcup\limits_n} \, B_n} = \overline{B_1 \cup B_2 \cup \cdots}.$$

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 $$x \in {\textstyle\bigcap\limits_n} \, \overline{A_n} = \overline{A_1} \cap \overline{A_2} \cap \cdots,$$ 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.