Fréchet–Urysohn space

Remarks

A Fréchet–Urysohn space is sequential: if $A$ is sequentially closed and $x \in \overline{A}$, a sequence in $A$ converges to $x$, so $x \in A$. Every subspace of a Fréchet–Urysohn space is Fréchet–Urysohn. A sequential space is Fréchet–Urysohn if and only if each of its subspaces is sequential.

Example: every first-countable space is Fréchet–Urysohn. For $x \in \overline{A}$ and a decreasing neighbourhood base $U_1 \supseteq U_2 \supseteq \cdots$ at $x$, pick $x_n \in A \cap U_n$; then $x_n \to x$.

Non-example: let $X$ be an uncountable set whose closed sets are $X$ and the countable subsets, and let $A \subset X$ with $A$ and $X \setminus A$ uncountable. Then $\overline{A} = X$, but the only convergent sequences are eventually constant. So a point $x \notin A$ is in $\overline{A}$ but is not the limit of a sequence in $A$.

The name Fréchet space is also used, but it has other meanings.