# nLab countably tight space

## Idea

The topology of a countably tight space is determined by countable subsets similar to how in a sequential space the topology is generated by sequences.

## Definition

A topological space $X$ is called countably tight or countably generated if a set $A \subset X$ is closed if (and hence only if) for each countable subset $D \subset X$ the intersection $A \cap D$ is closed in the relative topology on $D$.

## Categorical properties

The category of countably tight spaces is a coreflective subcategory of Top. More precisely, it is the coreflective hull of the subcategory of countable spaces.

Axioms: axiom of choice (AC), countable choice (CC).

### Implications

• a metric space has a $\sigma$-locally discrete base

• Nagata-Smirnov metrization theorem

• a second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of a countable cover of $X$.

• second-countable spaces are separable: use the axiom of countable choice to choose a point in each set of a countable cover.

• separable spaces satisfy the countable chain condition: given a dense set $D$ and a family $\{U_\alpha : \alpha \in A\}$, the map $D \cap \bigcup_{\alpha \in A} U_\alpha \to A$ assigning $d$ to the unique $\alpha \in A$ with $d \in U_\alpha$ is surjective.

• separable spaces are weakly Lindelöf: given a countable dense subset and an open cover choose for each point of the subset an open from the cover.

• Lindelöf spaces are trivially also weakly Lindelöf.

• a space with a $\sigma$-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a $\sigma$-locally finite base.

• a first-countable space is obviously Fréchet-Urysohn.

• a Fréchet-Uryson space is obviously sequential.

• a sequential space is obviously countably tight.

## References

Last revised on April 5, 2019 at 23:38:50. See the history of this page for a list of all contributions to it.