Fréchet–Urysohn space

Let X be a topological space. For any subset S of X, the sequential closure of S is the set

${\displaystyle \operatorname {SeqCl} _{X}S:=\left\{s\in X:{\text{ there exists a sequence }}\left(s_{i}\right)_{i=1}^{\infty }{\text{ in }}S{\text{ such that }}\left(s_{i}\right)_{i=1}^{\infty }\to x{\text{ in }}X\right\}}$.

A space X is said to be a Fréchet–Urysohn space if for every subset subset S of X, ${\displaystyle \operatorname {SeqCl} _{X}S=\operatorname {Cl} _{X}S}$, where ${\displaystyle \operatorname {Cl} _{X}S}$ denotes the closure of S in X.

If S is any subset of X then:

• a sequence ${\displaystyle x_{1},x_{2},\ldots }$ is eventually in S if there exists an positive integer N such that ${\displaystyle x_{n}\in S}$ for all ${\displaystyle n\geq N}$;
• S is sequentially open if each sequence (x_n) in X converging to a point of S is eventually in S;
  • Typically, if X is understood then ${\displaystyle \operatorname {SeqCl} S}$ is written in place of ${\displaystyle \operatorname {SeqCl} _{X}S}$.
• S is sequentially closed if ${\displaystyle S=\operatorname {SeqCl} _{X}S}$, or equivalently, if whenever (x_n) 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. Every open subset of X is sequentially open and every closed set is sequentially closed. The converses are not generally true. The spaces for which the converse is true are called sequential spaces; that is, a sequential space is a topological space in which every sequentially open subset is necessarily open (or equivalently, a space in which every sequentially closed subset is necessarily closed). Every Fréchet-Urysohn space is a sequential space but there are sequential spaces that are not Fréchet-Urysohn spaces.

Sequential (resp. Fréchet-Urysohn) spaces can be viewed as exactly those spaces X where for all subsets ${\displaystyle S\subseteq X}$, knowledge of which sequences in S converge to which point(s) of X is sufficient to determine whether or not S is closed in X (resp. to determine S’s closure in X).

Let ${\displaystyle \operatorname {SeqOpen} \left(X,\tau \right)}$ denote the set of all sequentially open subsets of the topological space ${\displaystyle \left(X,\tau \right)}$. Then ${\displaystyle \operatorname {SeqOpen} \left(X,\tau \right)}$ is a topology on X that contains the original topology ${\displaystyle \tau }$ (i.e. ${\displaystyle \tau \subseteq \operatorname {SeqOpen} \left(X,\tau \right)}$).

• Every Fréchet–Urysohn space is a sequential space.
  • The opposite implication is not true in general.[1][2]

• Every first-countable space is a Fréchet–Urysohn space.

A topological space ${\displaystyle X}$ is a strong Fréchet–Urysohn space 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 points ${\displaystyle x_{1}\in A_{1},x_{2}\in A_{2},\ldots }$ such that ${\displaystyle \{x_{n}\}\to x}$.

The above properties can be expressed as selection principles.

• Axioms of countability
• First-countable space
• Sequential space

• Arkhangel'skii, A.V. and Pontryagin, L.S., General Topology I, Springer-Verlag, New York (1990) ISBN 3-540-18178-4.
• Booth, P.I. and Tillotson, A., Monoidal closed, cartesian closed and convenient categories of topological spaces Pacific J. Math., 88 (1980) pp. 35–53.
• Engelking, R., General Topology, Heldermann, Berlin (1989). Revised and completed edition.
• Franklin, S. P., "Spaces in Which Sequences Suffice", Fund. Math. 57 (1965), 107-115.
• Franklin, S. P., "Spaces in Which Sequences Suffice II", Fund. Math. 61 (1967), 51-56.
• Goreham, Anthony, "Sequential Convergence in Topological Spaces"
• Steenrod, N.E., A convenient category of topological spaces, Michigan Math. J., 14 (1967), 133-152.