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 ⊆ 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 don't) 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, τ) be a topological space.

The sequential closure of a set S in X is the set:

SeqCl S  :=  [ S ]seq  :=  {  x ∈ X  :  there exists a sequence s• = (si)∞
i=1 in S such that s• → x in (X, τ) }

where  SeqClX S  or  SeqCl(X, τ) S  may be written if clarity is needed.

A space (X, τ) is said to be a Fréchet–Urysohn space if for every subset subset S of X, ClX S  =  SeqClX S, where denotes the closure of S in X.

Sequentially open/closed sets

Definitions: If S is any subset of X then:

  • a sequence x1,  x2,  ...  is eventually in S if there exists an positive integer N such that xn ∈ S for all integers n ≥ N.
  • S is sequentially open if each sequence (xn) in X converging to a point of S is eventually in S;
    • Typically, if X is understood then SeqCl S is written in place of SeqClX S.
  • S is sequentially closed if S  =  SeqClX S, or equivalently, if whenever x• = (xi)i ∈ I 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  SeqOpen(X, τ)  denote the set of all sequentially open subsets of the topological space (X, τ). The set  SeqOpen(X, τ)  is a topology on X that contains the original topology τ (i.e. τ ⊆ SeqOpen(X, τ)).

Strong Fréchet–Urysohn space

A topological space X is a strong Fréchet–Urysohn space if for every point x ∈ X and every sequence A1, A2, ... of subsets of the space X such that , there exist points a1 ∈ A1,  a2 ∈ A2,  ...  such that  (ai)∞
i=1 → x  in (X, τ).

The above properties can be expressed as selection principles.

Contrast to sequential spaces

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 any single given subset S ⊆ X, knowledge of which sequences in X converge to which point(s) of X (and which don't) is sufficient to determine whether or not S is closed in X (resp. to determine the closure of S in X).[note 1] Thus sequential spaces are those spaces X for which sequences in X can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in X; or said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence. In any space that is not sequential, there exists a subset for which this "test" gives a "false positive."[note 2]

Characterizations

Let (X, τ) be a topological space. Then the following are equivalent:

  1. X is a Fréchet–Urysohn space;
  2. For every subset S ⊆ X, SeqClX S = ClX S;
  3. Every subspace of X is a sequential space;
  4. For any subset S ⊆ X that is not closed in X and for every x ∈ (Cl S)  ∖  S, there exists a sequence in S that converges to x.
    • Contrast this condition to the following characterization of a sequential space:
    For any subset S ⊆ X that is not closed in X, there exists some x ∈ (Cl S)  ∖  S for which there exists a sequence in S that converges to x.[1]
    • This makes characterization implies that every Fréchet–Urysohn space is a sequential space.

Examples

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

Properties

Every Fréchet–Urysohn space is a sequential space. The opposite implication is not true in general.[2][3]

See also

Notes
  1. Of course, if you could use this knowledge to determine all of the sets in { T : S ⊂ T ⊆ X  } that are closed then you could determine the closure of S. This interpretation assumes that you make this determination only to the given set S and not to other sets; said differently, you cannot simultaneously apply this "test" to infinitely many subsets (e.g. you can't use something akin to the axiom of choice). It is in Fréchet-Urysohn spaces that the closure of a set S can be determined without it ever being necessary to consider any set other than S.
  2. Although this "test" (which attempts to answer "is this set open (resp. closed)?") could potentially give a "false positive," it can never give a "false negative;" this is because every open (resp. closed) subset S is necessarily sequentially open (resp. sequentially closed) so this "test" will never indicate "false" for any set S that really is open (resp. closed).

References
  1. Arkhangel'skii, A.V. and Pontryagin L.S.,  General Topology I, definition 9 p.12
  2. Engelking 1989, Example 1.6.18
  3. Ma, Dan. "A note about the Arens' space". Retrieved 1 August 2013.
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