Fréchet–Urysohn space


In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset, 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.
The space is named after Maurice Fréchet and Pavel Urysohn.

Definitions

Let X be a topological space.
For any subset S of X, the sequential closure of S is the set
A space X is said to be a Fréchet–Urysohn space if for every subset subset S of X,, where denotes the closure of S in X.
If S is any subset of X then:
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.
Every Fréchet-Urysohn space is a sequential space but there are sequential spaces that are not Fréchet-Urysohn spaces.
Sequential spaces can be viewed as exactly those spaces X where for all subsets, knowledge of which sequences in S converge to which point of X is sufficient to determine whether or not S is closed in X.
Let denote the set of all sequentially open subsets of the topological space.
Then is a topology on X that contains the original topology .

Characterizations

Let be a topological space.
Then the following are equivalent:
  1. X is a Fréchet–Urysohn space;
  1. every subspace of X is a sequential space.

    Properties

A topological space is a strong Fréchet–Urysohn space if for every point and every sequence of subsets of the space such that , there are points such that.
The above properties can be expressed as selection principles.