MINIMAL SEQUENTIAL HAUSDORFF SPACES

A sequential space (X,T) is called minimal sequential if no sequential topology on X is strictly weaker than T . This paper begins the study of minimal sequential Hausdorff spaces. Characterizations of minimal sequential Hausdorff spaces are obtained using filter bases, sequences, and functions satisfying certain graph conditions. Relationships between this class of spaces and other classes of spaces, for example, minimal Hausdorff spaces, countably compact spaces, H-closed spaces, SQ-closed spaces, and subspaces of minimal sequential spaces, are investigated. While the property of being sequential is not (in general) preserved by products, some information is provided on the question of when the product of minimal sequential spaces is minimal sequential.


Introduction.
All hypothesized spaces are Hausdorff topological spaces.If (X, T ) is a space and Q ⊂ X, {x ∈ X : some sequence in Q converges to x}, that is, the Tsequential closure of Q will be denoted by ∆ T (Q) (1) the collection of T -sequentially open subsets, which we denote by S T , is a topology on X, (2) T ⊂ S T , and (3) Q ∈ S T if and only if each sequence in X which is T -convergent to an element of Q is ultimately in Q.The space (X, T ) is called sequential if T = S T .In this case, the phrase "T is sequential" is often used.It is obvious that first countable spaces are sequential and it is known that a sequential space might fail to be first countable [4].In this paper, a space (X, T ) is minimal sequential if T is sequential and no sequential topology on X is strictly weaker (= smaller) than T .Such a space will be called minimal Hausdorff (sq).This terminology parallels the following from [1].
If P is a property of topological spaces, then P (1) will mean a space which is first countable and has property P, thus a space is Hausdorff (1) provided it is Hausdorff and first countable.It is proved in [4] that sequential spaces are characterized as quotients of first countable (metric) spaces.Hence, the class of minimal Hausdorff (sq) spaces coincides with the class of spaces which are minimal in the class of quotients of first countable (metric) spaces.Two proper subclasses of this class of spaces have been recently investigated in [9].
In Section 2, a number of characterizations of minimal Hausdorff (sq) spaces are established in terms of filter bases, sequences, and functions into such spaces satisfying certain graph conditions.These characterizations include parallels of those of minimal Hausdorff spaces by Bourbaki [2] in terms of open filter bases, and of those by Herrington and Long [6] in terms of arbitrary filter bases.These characterizations reveal that a number of spaces which have been the object of study are minimal Hausdorff (sq).Indeed, minimal P spaces, where P is either of the properties semimetrizable, symmetrizable, neighborhood Ᏺ, Ᏺ, weakly first countable, are minimal Hausdorff (sq) (see [11,12]).In Section 3, some relationships between the class of minimal Hausdorff (sq) spaces and other classes of spaces, for example, minimal Hausdorff spaces, countably compact spaces, H-closed spaces, and SQ-closed spaces in the sense of Thompson [14], are determined.It is established that minimal Hausdorff sequential spaces, as well as sequentially compact sequential spaces, are minimal Hausdorff (sq) and that every minimal Hausdorff (sq) space is sequentially H-closed.Sequentially H-closed spaces were studied by Thompson [14] and Espelie et al. [3] under the name of SQ-closed spaces.Examples are provided to show that a minimal Hausdorff (sq) space need not be H-closed and that a sequential H-closed space might fail to be minimal Hausdorff (sq).Other examples are given to distinguish this class of spaces from other classes of spaces.In Section 3, subspaces and products of minimal Hausdorff (sq) spaces are studied.Parallel to the result of Katětov [10] and Stone [13] that a space is compact if and only if every closed subspace is H-closed, we prove that a space is countably compact if and only if every closed subspace is SQ-closed.Although the property of a space being minimal Hausdorff (sq) is not preserved by products, we apply results from [3,5] to provide some information on products of such spaces in this section.

Characterizations of minimal Hausdorff (sq) spaces.
Preliminary to our first theorem, we introduce some additional concepts and notations which are utilized throughout the paper.If (X, T ) is a space and Q ⊂ X, we use the notation Σ T (Q) (Σ T (x) if Q = {x}) for the collection of elements of T which contain Q (simply Σ(Q) when confusion is unlikely); we let Q (Q • ) represent the closure (interior ) of Q.If Ω is a filter base on a space, adh Ω will represent its adherence (adh [3], and that X is Hausdorff if and only if [15].The notions θcluster point and θ-convergence of nets are similarly defined.If Ᏺ is a filter on a space (X, T ), the filter base T ∩ Ᏺ will be called the open part of Ᏺ and will be denoted by ᏻ(Ᏺ), {A ∈ ᏻ(Ᏺ) : A is regular open} will be called the regular-open part of Ᏺ and can be denoted by • , and a topological space is semiregular if the regular-open subsets of the space are a base for its topology.Since it is an elementary fact of General Topology that A • • = A • for any subset A of a topological space, we see that adh Theorem 2.1.The following statements are equivalent for a space (X, T ): a topology on X and T * ⊂ T , then S T = S T * ; (v) each countable filter base on X with at most one (T − θ)-adherent point T -converges; (vi) all topologies on X which are smaller than T have the same convergent sequences as T ; (vii) if Ᏺ is a filter on X with a countable base and ᏻ(Ᏺ) has at most one T -adherent point, then ᏻ(Ᏺ) converges; (viii) if Ᏺ is a filter on X with a countable base and (Ᏺ) has at most one T -adherent point, then (Ᏺ) converges. x.
If h is a subnet of x n and h → θ z with respect to T , then h → θ z with respect to T * since The proof is obvious.
(iv)⇒(v).Let N be the set of positive integers, let Ω = {F n : n ∈ N} be a filter base, and let v ∈ X such that F n+1 ⊂ F n , [Ω] θ ⊂ {v}, and suppose Ω does not converge.Choose a sequentially closed and this contradicts (iv).(v)⇒(vi).Let x n be a sequence in X and suppose x n → T * x. Using the same argument as in the proof of (i)⇒(ii) above, x is the only possible (T − θ)-adherent point of the elementary filter generated by x n .Hence, x n → T x.
(vi)⇒(i).Let x n be a sequence in X with at most one (T − θ)-cluster point.Without loss of generality, choose v ∈ X such that no cluster point of x n is in X −{v}.Employ the same construction as in the proof of (iv)⇒(v) to get a Hausdorff topology T * on X such that T * ⊂ T , and (v)⇒(vii).Let Ᏺ be a filter on X with countable base Ω, v ∈ X, and suppose adh ᏻ(Ᏺ) ⊂ {v}.
In view of Theorem 2.1, we have the following theorem.
Theorem 2.2.The following statements are equivalent for a sequential space (X, T ): In [14], Thompson introduced the class of SQ-closed spaces.A space is SQ-closed if its continuous image in any Hausdorff space is sequentially closed.In [3], it is proved that a space is SQ-closed if and only if every countable filter base on the space with at most one θ-adherent point θ-converges.
Corollary 2.4 is immediate in view of this result and equivalence (vi) of Theorem 2.2.

Corollary 2.4. A minimal Hausdorff (sq) space is SQ-closed. Corollary 2.5. A minimal Hausdorff (sq) space is H-closed if and only if it is minimal Hausdorff.
Next we present characterizations of minimal Hausdorff (sq) spaces in terms of functions into such spaces satisfying certain graph conditions.Let X, Y be spaces and let f : X → Y .We will say that f has a subclosed (strongly subclosed) graph if adh(f (Ω)) ⊂ {f (x)} ([f (Ω)] θ ⊂ {f (x)}) for each x ∈ X and filter base Ω on X − {x} satisfying Ω → x.When f has a subclosed (strongly subclosed) graph and {f (x)} is closed (θ-closed) in Y for each x ∈ X, we say that f has a closed (strongly closed) graph.The notion of strongly closed graph was introduced by Herrington and Long in [6], while subclosed and strongly subclosed graphs were introduced by Joseph [7].The function f is said to be sequentially continuous at x ∈ X if f (x n ) → f (x) for each sequence x n in X satisfying x n → x, and is said to be sequentially continuous if it is sequentially continuous at each x ∈ X.If X is a nonempty set, v ∈ X, and Ω is a filter base with empty intersection on X − {v}, X(v, Ω) denotes X with the topology Theorem 2.6.The following statements are equivalent for a sequential space Z: (i) Z is minimal Hausdorff (sq); (ii) for each space X, each f : X → Z with a strongly closed graph is sequentially continuous; (iii) for all spaces X, Y and f : X → Z, g : Y → Z with strongly closed and closed graphs, respectively, Ᏹ(f ,g,X × Y ,Z) is a sequentially closed subset of X × Y ; (iv) for each space X and all f ,g : X → Z with strongly closed and closed graphs, respectively, Ᏹ(f ,g,X,Z) is a sequentially closed subset of X; (v) for each space X and all f ,g : X → Z with strongly closed and closed graphs, respectively, Ᏹ(f ,g,X,Z) = X whenever ∆(Ᏹ(f ,g,X,Z)) = X.
Proof.(i)⇒(ii).Let X be a space, let f : X → Z have a strongly closed graph, let x ∈ X be a point which is not isolated, and let x n be a sequence in X − {x} such that x n → x.Let Ω be the usual base for the elementary filter generated by {x n }.Then [f (Ω)] θ = {f (x)}, since Z is minimal Hausdorff (sq), it follows by equivalence (vi) of Theorem 2.2 that f (Ω) → f (x) and the proof that f is sequentially continuous is complete.
(ii)⇒(iii).Suppose X, Y are spaces, f : X → Z, g : Y → Z are functions with strongly closed and closed graphs, respectively, and let (x n ,y n ) be a sequence in Ᏹ(f ,g,X × Y ,Z) such that (x n ,y n ) → (x, y).Then x n → x, y n → y, and f (x n ) → f (x) from (ii).Hence, g(y n ) → f (x) and g(y) = f (x) since g has a closed graph.Therefore, (x, y) ∈ Ᏹ(f ,g,X × Y ,Z).
(iii)⇒(iv).Let X be a space and let f ,g : X → Z have strongly closed and closed graphs, respectively.Then Ᏹ(f ,g,X,Z) = π(Ᏹ(f ,g,X × X, Z) ∩ Ᏸ), where π is the projection of X × X onto X and Ᏸ is the diagonal of X × X.From (iii), Ᏹ(f ,g,X × X, Z) ∩ Ᏸ is a sequentially closed subset of Ᏸ, and the restriction of π to Ᏸ is a homeomorphism, so (iv) holds.
To establish that f and g have strongly closed and closed graphs, respectively, we need to check only at v. If Λ is a filter base on Z − {v} and [14] has proved that every countably compact space is SQ-closed.Theorem 3.1 parallels the well-known result that a space is compact if and only if each of its closed subspaces is H-closed [10,13].

Relationships between the class of minimal Hausdorff (sq) spaces and the classes of countably compact spaces, minimal Hausdorff spaces, SQ-closed spaces, and H-closed spaces. Thompson
Theorem 3.1.A space X is countably compact if and only if each closed subspace of X is SQ-closed.
Proof.If (X, T ) is countably compact, then each closed subspace is countably compact, and hence is SQ-closed.Conversely, suppose that each closed subspace of X is SQ-closed and that A ⊂ X is infinite and has no limit points in X.Then A is a closed subspace of X which is discrete in its relative topology, and is therefore not SQ-closed.
We have the following result for minimal Hausdorff (sq) spaces.

Theorem 3.2. A sequentially compact sequential space is minimal Hausdorff (sq).
Proof.Let (X, T ) be a sequentially compact sequential space, let v ∈ X, and let x n be a sequence in X with no θ-cluster point in X −{v}.If V 0 ∈ Σ(v) and x n is frequently in X −V 0 , then some subsequence of x n converges to some point in X −V 0 , a contradiction to the assumption that x n has no θ-cluster point in X −{v}.Therefore x n → T v.

Corollary 3.3. A countably compact sequential space is minimal Hausdorff (sq).
Proof.A countably compact sequential space is sequentially compact.Corollary 3.4.If each closed subspace of a sequential space (X, T ) is SQ-closed, then X is minimal Hausdorff (sq).
In [6], Herrington and Long proved that a Hausdorff space is minimal Hausdorff if and only if each filter base on the space with at most one θ-adherent point converges.In view of this result, it is obvious that every sequential minimal Hausdorff space is minimal Hausdorff (sq).But, as can be seen from the following example, a minimal Hausdorff (sq) space need not be minimal Hausdorff.
Example 3.5.The space [0, Ω) of ordinals less than the first uncountable ordinal endowed with the order topology is not minimal Hausdorff although it is countably compact and first countable.Example 3.5 also establishes that a minimal Hausdorff (sq) space need not be Hclosed since the space there is regular and is not compact.
The following example shows that a first countable minimal Hausdorff space need not be countably compact.The space is the classical example of a countable minimal Hausdorff space which is not compact.
The next example shows that a sequential H-closed space need not be minimal Hausdorff (sq).Proof.If every closed subspace of a space is minimal Hausdorff (sq), then each closed subspace is SQ-closed and the space is countably compact from Theorem 3.1.Since a countably compact sequential space is sequentially compact, the necessity is established.The sufficiency follows from Theorem 3.2.
Proof.From [4], X µ is sequential.Now, let y n be a sequence in X µ and y ∈ X µ such that the filter base Ω defined by F n = {y k : k ≥ n} satisfies [Ω] θ ⊂ {y}.Choose v ∈ X and let v n be the point in X with µ-coordinate y n and every other coordinate the same as that of v. Then the point x ∈ X with µ-coordinate y and all other coordinates the same as those of v is the only possible θ-cluster point of the sequence v n .Hence, v n → x and y n → y.Theorem 3.10.If X µ is a family of H-closed minimal Hausdorff (sq) spaces such that the product X = ΠX µ is sequential, then X is minimal Hausdorff (sq).

Example 3 . 7 .Theorem 3 . 8 .
Let Y = {0} ∪ (N − {1}) ∪ {j + 1/2n : j, n ∈ N − {1}} with the subspace topology T from X in Example 3.6.Then 0 is the only θ-cluster point of x n defined byx n = n + 1, but x n → 0.It is well known that both compactness and sequential compactness imply countable compactness.We have the following implication diagram for the class of sequential spaces; none of the implications is reversible: Every closed subspace of a sequential space is minimal Hausdorff (sq) if and only if the space is sequentially compact.
a topology on X and T * ⊂ T , then S T = S T * ; (vi) each countable filter base on X with at most one (T − θ)-adherent point T -converges; (vii) all topologies on X which are smaller than T have the same convergent sequences as T ; (viii) if Ᏺ is a filter on X with a countable base and ᏻ(Ᏺ) has at most one T -adherent point, then ᏻ(Ᏺ) converges; (ix) if Ᏺ is a filter on X with a countable base and (Ᏺ) has at most one T -adherent point, then (Ᏺ) converges.Corollary 2.3 comes easily from equivalence (ix) of Theorem 2.2.Corollary 2.3.A minimal Hausdorff (sq) space is semiregular.