Countable tightness

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) by the convergent sequences.

The countable generated spaces are precisely the spaces having countable tightness - therefore the name countably tight is used as well.

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

A topological space ${\displaystyle X}$ has countable fan tightness if for every point ${\displaystyle x\in X}$ and every sequence ${\displaystyle A_{1},A_{2},\ldots }$ of subsets of the space ${\displaystyle X}$ such that ${\displaystyle x\in \bigcap _{n}{\overline {A_{n}}}}$ , there are finite set ${\displaystyle B_{1}\subset A_{1},B_{2}\subset A_{2},\ldots }$ such that ${\displaystyle x\in {\overline {\bigcup _{n}B_{n}}}}$.

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