NOTES ON FRÉCHET SPACES

First, we introduce sequential convergence structures and characterize Fréchet spaces and continuous functions in Fréchet spaces using these structures. Second, we give sufficient conditions for the expansion of a topological space by the sequential closure operator to be a Fréchet space and also a sufficient condition for a simple expansion of a topological space to be Fréchet. Finally, we study on a sufficient condition that a sequential space be Fréchet, a weakly first countable space be first countable, and a symmetrizable space be semi-metrizable.


Introduction.
Let us recall the following definitions in a topological space (X, c) endowed with the topological closure operator c.We denote the set of all positive integers by N.
(a) Fréchet [3] (also called Fréchet-Urysohn [2]): for each subset A of X, c(A) = {x ∈ X | (x n ) converges to x for some sequence (x n ) of points in A}.
(b) sequential [5]: for every subset A of X which is not closed in X, there exists a sequence (x n ) of points in A converging to a point of the set c(A) − A.
(c) weakly first countable [4] (also called g-first countable [1] and [13]): for each x ∈ X, there exists a family {B(x, n) | n ∈ N} of subsets of X such that the following conditions are satisfied: (i) x ∈ B(x, n + 1) ⊂ B(x, n) for all n ∈ N, (ii) a subset U of X is open if and only if for every x ∈ U there exists an n ∈ N such that B(x, n) ⊂ U .Such a family {B(x, n) | n ∈ N} is called a weak base at x.(d) Symmetrizable [8]: there exists a symmetric (= a metric except for the triangle inequality) d on X satisfying the following condition: a subset U of X is open if and only if for every x ∈ U there is a positive real number r such that B(x, r ) ⊂ U , where B(x, r ) denotes the set {y ∈ X | d(x, y) < r }.
(e) Semi-metrizable [7]: there exists a symmetric d on X such that for each x ∈ X, the family {B(x, r ) | r > 0} forms a (not necessarily open) neighborhood base at x.
Many topologists have studied on properties of spaces mentioned above and relationships among the spaces.In particular, S. P. Franklin in [6, Props.7.2 and 7.3] gave sufficient conditions for sequential spaces to be Fréchet and the following theorems were obtained by G. Gruenhage in [7] and F. Siwiec in [13].
Theorem B [13,Thm. 1.10].If a Hausdorff space X satisfies one of the properties in the second column of the above diagram and is also Fréchet, then it satisfies the corresponding property in the first column.
The purpose of this paper is to consolidate and investigate properties of Fréchet spaces.First, we introduce sequential convergence structures and characterize Fréchet spaces and continuous functions in Fréchet spaces using these structures.Second, we give sufficient conditions for the expansion of a topological space by the sequential closure operator to be a Fréchet space and also a sufficient condition for a simple expansion of a topological space to be Fréchet.Finally, we study on a sufficient condition that a sequential space be Fréchet, a weakly first countable space be first countable, and a symmetrizable space be semi-metrizable.And then we obtain generalizations of Theorems A and B.
Standard notations, not explained below, is the same as in [3,14].

Sequential convergence structures.
Let X be a nonempty set and let S(X) denote the set of all sequences of points in X. Recall that a nonempty subfamily L of the cartesian product S(X)×X is called a sequential convergence structure on X [9] if it satisfies the following conditions: (SC 1): For each x ∈ X, ((x), x) ∈ L, where (x) is the constant sequence whose nth term is x for all indices n ∈ N.
Let SC[X] denote the set of all sequential convergence structures on X.
Theorem 2.1 [9].For L ∈ SC[X], define a function c L : P (X) → P (X) as follows: for is a Fréchet space endowed with the topological closure operator c L .
Let ᏸ(c) denote the set of all pairs ((x n ), x) ∈ S(X) × X such that (x n ) converges to x in a topological space (X, c) endowed with the topological closure operator c.Theorem 2.2 [9].
Note that there is an L ∈ SC[X] such that L ≠ ᏸ(c L ).Let Q be the rational number space with the usual topology and let ) is either strictly increasing or strictly decreasing}.Then we have According to Theorem 2.1 and Theorem 2.2, we immediately obtain the following theorem and hence we omit the proof.
Now, we characterize continuous functions in Fréchet spaces using sequential convergence structures.
Proof.Let ((x n ), x) ∈ L X .Then, by Theorem 2.2(1) and hypothesis, (( Conversely, suppose that there is a closed subset Proof.Since (X, c X ) is a Fréchet space, c x = c ᏸ(c X ) , and hence we can replace L X by ᏸ(c X ) in Theorem 2.4 above.
Remark.Note that we thus obtain by Theorem 2.4 a convenient method to check whether a function in Fréchet spaces is continuous or not.For example, let f be a realvalued function defined on a subspace X of the real line R with the usual topology and let is either strictly increasing or strictly decreasing}.Then, it is obvious that L X ∈ SC[X] and moreover (X, c L X ) is precisely equal to the space X itself.By Theorem 2.4, we see that f is continuous if and only if for each ((x n ), x) ∈ L X ,(f (x n )) converges to f (x) in R.

Expansions.
Let (X, c) be a topological space endowed with the topological closure operator c.A function c * : P (X) → P (X) defined by for each A ⊂ X, c * (A) = {x ∈ X | (x n ) converges to x in the space X for some (x n ) ∈ S(A)} is called the sequential closure operator [3] on the topological space (X, c).It is obvious that a topological space (X, c) is Fréchet (sequential) if and only if for each It is known that the sequential closure operator c * on a topological space (X, c) need not be a topological closure operator on the set X and if c * is idempotent, then (X, c * ) is a Fréchet expansion of (X, c).
Consider the following two properties in a topological space (X, c): We then have easily that the following implications hold: Fréchet ⇒ ( * ) ⇒ ( * * ).But, according to [10, Ex. 2.4], we see that the converses are not true.Note that ( * * ) is equivalent to (SC 3) of ᏸ(c).

Remark.
It is an interesting and important fact that ᏸ(c) = ᏸ(c * ), even though c ≠ c * .From this fact, we have naturally the following: (1) The properties ( * ) and ( * * ) are sufficient conditions for a non-Fréchet space (X, c) to have the Fréchet expansion (X, c * ).
(2) There are close corelations between some topological properties of the two spaces (X, c) and (X, c * ).For example, (i) the separation properties of (X, c) transfer to the space (X, c * ), (ii) if (X, c * ) is compact (connected or separable), then (X, c) is compact (resp.connected or separable), and (iii) (X, c) is sequentially compact if and only if (X, c * ) is sequentially compact, etc.Let (X, -) be a topological space and Ꮽ a family of subsets of X.The expansion of by Ꮽ denoted by -(Ꮽ) is the topology on X with -∪Ꮽ as sub-base.In case Ꮽ = {A} the expansion by Ꮽ is a simple expansion, denoted by -(A).Simple extensions (expansions) were introduced by N. Levine in [11].This and subsequent works have been concerned with the preservation of topological properties under these expansions.In particular, J. A. Narvarte and J. A. Guthrie [12] investigated the preservation of Fréchet spaces, sequential spaces and k-spaces under simple expansions.
We recall that a family Ꮽ of subsets of X is point finite [14] if and only if each x ∈ X belongs to only finitely many A ∈ Ꮽ. Proof.If (X, -) is a Fréchet space, then clearly Ꮽ = ∅, and so -(Ꮽ) = -.Hence, it remains to prove the case that (X, -) . Hence, we have that there exists (x n ) ∈ S(Y ∩ M) such that (x n ) converges to p in (X, -).To end the proof, we claim that (x n ) converges to p in (X, -(Ꮽ)).Suppose that it is not.Then there exists a basic open set U in (X, -(Ꮽ)) containing p such that (x n ) is not eventually in U. We have already known that V ∩ M ⊂ U for some open set V in (X, -) containing p.It follows that there is an open set V in (X, -) containing p such that (x n ) is not eventually in V ∩ M, and hence (x n ) is also not eventually in V because (x n ) ∈ S(M), which is a contradiction.

Fréchet and related spaces.
We start with the following property in a topological space (X, c) endowed with the topological closure operator c: Obviously, every Hausdorff Fréchet space satisfies the property ( * * * ).On the other hand, there are Hausdorff compact spaces satisfying the property ( * * * ) which are not sequential and hence are not Fréchet.
Example 2.8.The ordinal space X = [0,ω 1 ], where ω 1 is the first uncountable ordinal, is a Hausdorff compact space which is not sequential (see [10, Ex. 2.4(2)]).Now we show that X satisfies the property ( * * * ).Since [0,ω 1 ) is discrete, to show this it is sufficient to prove that for each A ⊂ [0,ω 1 ), if ω 1 ∈ c(A) then there exists a subset B of A such that ω 1 ∈ c(B) and c(B) = B ∪ {ω 1 }.If there exists α ∈ [0,ω 1 ) such that [α, ω 1 ) ⊂ A, then [α, ω 1 ) is a desired subset of A. Hence, it remains to prove the case where for each α ∈ [0,ω 1 ), (α, ω 1 ) A. We first recall a definition that a subset C of [0,ω 1 ] is an interval if and only if for each α, β ∈ C with α β, [α, β] ⊂ C.Then, it is clear that A is the union of infinitely many intervals: A = ∪{C i | i ∈ I}, where C i is an interval for each i ∈ I and C i ∩ C j = ∅ if i ≠ j.Let α i denote the first element of C i for each i ∈ I.Note that I must be uncountable.Suppose on the contrary that I is countable.Then, let 1 0 be the first element of I and assume α 1 0 ≠ 0,(α i − 1 | i ∈ I) ∈ S([0,ω 1 ) − A), and hence sup{α i − 1 | i ∈ I} ω 1 since there (3) From Theorems 2.9, 2.10 and Corollary 2.12, we know that the property ( * * * ) is a sufficient condition that a topological space satisfying one of the properties in the second column of the above diagram given in the introduction satisfy the corresponding property in the first column.This is a generalization of Theorem B [13,Thm. 1.10].

Theorem 2 . 4 .
Let (X, c X ) and (Y , c Y ) be two Fréchet spaces endowed with the topological closure operators c X and c Y , respectively, and let L

Corollary 2 . 5 .
which is a contradiction.By Theorem 2.4, we have the following well-known fact.Let (X, c X ) and (Y , c Y ) be two Fréchet spaces.Then, a function f : (X, c X ) → (Y , c Y ) is continuous if and only if f is sequentially continuous.
2,...,n}.Then, clearly, Y ∩ M ≠ ∅ since M is a basic open set in (X, -(Ꮽ)) containing p.We first show that p ∈ c -(Ꮽ) (Y ∩ M).Since -∪ Ꮽ is a sub-base for -(Ꮽ), by the definition of M, we have that for each basic open set U in (X, -(Ꮽ)) containing p,(∩{V j | i ∈ J}) ∩ M ⊂ U for some finite family {V j | j ∈ J and J is finite} of open sets V j in (X, -) containing p, and so V ∩ M ⊂ U for some open set V in (X, -) containing p. Hence, it is sufficient to show that for each open set V in (X, -) containing p, (Y ∩ M)∩ V ≠ ∅.Suppose on the contrary that there exists an open