Fréchet–Urysohn space

In the field of topology, a Fréchet–Urysohn space is a topological space $X$ with the property that for every subset $S\subseteq X$ the closure of $S$ in $X$ is identical to the sequential closure of $S$ in $X.$ Fréchet–Urysohn spaces are a special type of sequential space.

Fréchet–Urysohn spaces are the most general class of spaces for which sequences suffice to determine all topological properties of subsets of the space. That is, Fréchet–Urysohn spaces are exactly those spaces for which knowledge of which sequences converge to which limits (and which sequences do not) suffices to completely determine the space's topology. Every Fréchet–Urysohn space is a sequential space but not conversely.

The space is named after Maurice Fréchet and Pavel Urysohn.

Definitions

See also: Sequential space

Let $(X,\tau )$ be a topological space. The sequential closure of $S$ in $(X,\tau )$ is the set:

$${\begin{alignedat}{4}\operatorname {scl} S:&=[S]_{\operatorname {seq} }:=\left\{x\in X~:~{\text{ there exists a sequence }}s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }\subseteq S{\text{ in }}S{\text{ such that }}s_{\bullet }\to x{\text{ in }}(X,\tau )\right\}\end{alignedat}}$$

where $\operatorname {scl} _{X}S$ or $\operatorname {scl} _{(X,\tau )}S$ may be written if clarity is needed.

A topological space $(X,\tau )$ is said to be a Fréchet–Urysohn space if

$$\operatorname {cl} _{X}S=\operatorname {scl} _{X}S$$

for every subset $S\subseteq X,$ where $\operatorname {cl} _{X}S$ denotes the closure of $S$ in $(X,\tau ).$

Sequentially open/closed sets

Suppose that $S\subseteq X$ is any subset of $X.$ A sequence $x_{1},x_{2},\ldots$ is eventually in $S$ if there exists a positive integer $N$ such that $x_{i}\in S$ for all indices $i\geq N.$

The set $S$ is called sequentially open if every sequence $\left(x_{i}\right)_{i=1}^{\infty }$ in $X$ that converges to a point of $S$ is eventually in $S$; Typically, if $X$ is understood then $\operatorname {scl} S$ is written in place of $\operatorname {scl} _{X}S.$

The set $S$ is called sequentially closed if $S=\operatorname {scl} _{X}S,$ or equivalently, if whenever $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ is a sequence in $S$ converging to $x,$ then $x$ must also be in $S.$ The complement of a sequentially open set is a sequentially closed set, and vice versa.

Let

$$\begin{array}{rl}\mathrm{SeqOpen}<!--\; \; -->(X,\mathrm{\tau <!--\; \tau \; -->}):& =\left\{S\subseteq <!--\; \subseteq \; -->X:S\text{ is sequentially open in }(X,\mathrm{\tau <!--\; \tau \; -->})\right\}\\ & =\{S\subseteq <!--\; \subseteq \; -->X:S={\mathrm{SeqInt}}_{(X,\mathrm{\tau <!--\; \tau \; -->})}\end{array}$$