On the 𝑯 -space and the product of two 𝑯 -spaces

Considering the set of real numbers R, for each x ∈ A, B(x)={(x−ϵ,x+ϵ):ϵ>0}, and for each x∈ R\A, let B(x)={[x,x+ϵ):ϵ>0}, the unique topology generated by {B(x): x ∈ R} is denoted by τ(A) and (R, τ(A)) is called an H -space. In this paper, we give some results about these spaces and the product of two of them, including the separation axioms, w D property, various types of compactness and connectedness, and weaker properties of normality.


Introduction
defined on the set of real numbers ℝ topologies that lie between the usual topology and the Sorgenfrey topology; ℝ with these topologies were called -spaces. These spaces were previously studied in Bouziad and Sukhacheva (2017), Chatyrko and Hattori (2016;2013), and Kulesza (2017). In this paper, we give some results about these spaces and the product of two of them. Throughout this paper, we denote the set of positive integers by ℕ, the rationals by ℚ, the irrationals by ℙ, and the set of real numbers by ℝ. A 4 space is a 1 normal space and a Tychonoff ( 3 1 2 ) space is a 1 completely regular space. We do not assume 2 in the definition of compactness, paracompactness and countable compactness. We do not assume regularity in the definition of Lindelöfness. For a subset of a space , i and denote the interior and the closure of , respectively.
Observe that if we interchange the local bases in Definition 1.1 and define a new topology on ℝ, then for any subset of ℝ, we have = (ℝ\ ). Note that if = ∅, then ( ) = and if = ℝ, then ( ) = . From now on, when we consider antopology ( ) on ℝ, we are assuming that is a nonempty proper subset of ℝ. Observe that if ∈ ℝ\ and [ , ) ∈ ℬ( ), then [ , ) need not be clopen (closed-and-open) because could be an element of . But if , ∈ ℝ\ , then [ , ) is clopen. It is clear that for any subset of ℝ we have that is coarser than ( ) and ( ) is coarser than , i.e., ⊂ ( ) ⊂ , (Hattori, 2010). So, it is clear that every -space is completely Hausdorff. Any -space is first countable, hence Fréchet, sequential, and of countable tightness, (Engelking, 1977).
Claim 1: The set = \ is a subset of ℝ\ .
Recall that a space is called if is 4 and × is not normal where = [0,1] is considered with its usual metric topology. Since the product of a Lindelöf space with a compact space is Lindelöf (Engelking, 1977), then for any subset of ℝ, ( ℝ , ( ) ) × ( , ) is Lindelöf. Since the product is also 3 , then it is normal. Thus we conclude the following theorem.

Theorem 1.4: Every -space (ℝ, ( )) is not Dowker.
Recall that a space is said to satisfy property (Nyikos, 1981) if for every infinite closed discrete subspace of , there exists a discrete family { : ∈ ℕ } of open subsets of such that each meets at exactly one point. That is, for each ∈ ℕ we have ∩ is a singleton.
Observe that {(− , ): ∈ ℕ} is an open cover for (ℝ, ( )) that has no finite subcover, so (ℝ, ( )) is not compact nor countably compact. Recall that a space is pseudo compact if is a Tychonoff and every continuous real-valued function defined on is bounded. In a 4 space, this is equivalent to countable compactness (Engelking, 1977). Hence, for any choice of ⊂ ℝ the -space ( ℝ , ( ) ) is neither compact, countably compact, nor pseudo compact.
Let us consider the compact subsets in anyspace, ( ). By the relation ⊂ ( ) ⊂ , we know that any compact subset of ( ) must be compact in , so it is closed and bounded (in the sense of the usual metric). Moreover, any subset that is compact in must be compact in ( ), and these are exactly the bounded -closed subsets that contain no strictly increasing sequence (Espelie and Joseph, 1976). Hence, it is natural to ask for a characterization of compact subsets of ( ). Such characterization must depend on , of course.
It is clear that a subset of is compact in ( ) if and only if it is compact in , and a subset of ℝ\ is compact in ( ) if and only if it is compact in . By taking finite unions of compact subsets of and ℝ\ , one can trivially construct some compact subsets of ( ). While we are far from having a complete characterization of compact subsets of an -space, we give an example of a compact subset that cannot be constructed in this way. . This is a compact subset. Note that ∩ is compact in , yet ∩ ℝ\ is not compact in , as it is not closed.
A space is totally imperfect if every compact subspace of is countable. For example, the Sorgenfrey line is totally imperfect, while the real line is not. The following was proved in Bouziad and Sukhacheva (2017): "Space (ℝ, ( )) is totally imperfect if and only if is totally imperfect". This leads to the following results.
Recall that a space is sequentially compact if is Hausdorff and any sequence of elements of has a convergent subsequence. Observe that since ⊂ ( ) for any ∅ = ⊂ ℝ and both are Hausdorff, then if a sequence ( ) ∈ℕ converges to in (ℝ, ( )), then ( ) ∈ℕ also converges to in (ℝ, ). Thus if → in (ℝ, ), then → also in (ℝ, ( )). Any -space (ℝ, ( )) is not sequentially compact, and here is a counterexample.
Following Corollary 1.7, we have the next result.
When it comes to local compactness, it was also proved in Bouziad and Sukhacheva (2017) that " The -space ( ℝ , ( ) ) is locally compact if and only if ℝ\ is closed in (ℝ, ) and discrete in (ℝ, )". This can be generalized to a finite product. However, it is important to note that this is not the case for an infinite product of -spaces. By Engelking (1977), Theorem 3.3.13, we can deduce the following.
Recall that a space is said to be -normal if it is 1 and for any pair of disjoint closed subsets, and of there exist disjoint open subsets and such that ∩ is dense in and ∩ is dense in , (Ludwig, 2002). Clearly, 4 implies -normality. In Ludwig (2002), it was stated that "If is an -normal space, then any two disjoint closed discrete subsets of can be separated by two disjoint open subsets of ". Hence, we have the following statement.

Theorem 2.9: For any subsets, and of ℝ, (ℝ, ( )) × (ℝ, ( )) is paracompact if and only if it is Lindelöf.
The question "under what condition is the product of two normal spaces normal?" has been studied by many topologists. Partial answers to this question have been found. We apply some of those results to give explore when space (ℝ, ( )) × (ℝ, ( )) is normal. By Theorem 2.1, we know that if and are subsets of ℝ with countable complements, then (ℝ, ( )) × (ℝ, ( )) is paracompact (and normal). However, observing that the product of a metrizable space and a perfectly normal space is paracompact and perfectly normal (Michael (1953), Proposition 5), we actually have the following "strengthening" of this statement.
Theorem 2.10: If and are subsets of ℝ with ℝ\ countable, then (ℝ, ( )) × (ℝ, ( )) is perfectly normal and paracompact. Morita (1963) proved that "the product of a paracompact space which is a countable union of locally compact closed subsets, and a paracompact space is paracompact." Together with the paracompactness of every -space we may deduce the following result.
On the other hand, in Bouziad and Sukhacheva (2017), it was proved that "The -space ( ℝ , ( ) ) is locally compact if and only if ℝ\ is closed in (ℝ, ) and discrete in (ℝ, )." Thus, we get the following results.
Recall that a topological space is called normal if there exists a normal space and a bijective function : ⟶ such that the restriction ↾ : ⟶ ( ) is a homeomorphism for each Lindelöf subspace ⊆ (Kalantan and Saeed, 2017). In [12, Theorem 1.6], it was proved that "If is 3 separable -normal and of countable tightness, then is 4 .". Note that ( ℝ , ( ) ) × ( ℝ , ( ) ) is of countable tightness, 3 and separable. Hence, we have the following theorem. Recall that a topological space is callednormal if there exists a normal space and a bijective function : ⟶ such that the restriction ↾ : ⟶ ( ) is a homeomorphism for each separable subspace ⊆ (Kalantan and Alhomieyed, 2018). Since (ℝ, ( )) × (ℝ , ( )) is separable, the following is clear. Recall that a topological space is callednormal if there exists a normal space and a bijective function : ⟶ such that the restriction ↾ : ⟶ ( ) is a homeomorphism for each compact subspace ⊆ , (AlZahrani and . If is paracompact, is called 2paracompact, (Mohammed et al., 2019). For any subsets and of ℝ, we always have ( ℝ , ( ) ) × ( ℝ , ( ) ) is submetrizable. Since every submetrizable space is 2 -paracompact (Mohammed et al., 2019), we deduce the following result.
Recall that a topological space is called normal, (Kalantan and Alhomieyed, 2017), if there exists a normal space and a bijective function : ⟶ such that the restriction ↾ : ⟶ ( ) is a homeomorphism for each countably compact subspace ⊆ . We have the following theorem.
Proof: Consider ℝ 2 with its usual metric topology. Consider : (ℝ, 1 ) × (ℝ, 2 ) ⟶ ( ℝ 2 , ). It is continuous because the usual metric is coarser than the product of 1 and 2 . Let be any countably compact subspace of (ℝ, 1 ) × (ℝ, 2 ). By continuity, is countably compact in the metric space ( ℝ 2 , ), hence is closed in ( ℝ 2 , ). For the same reason : ( , 1 × 2 ) ⟶ ( , ) is a homeomorphism as it is a bijection, continuous and closed: any closed subset of in 1 × 2 is countably compact (countable compactness is hereditary with respect to closed subspaces), thus ( ) = is countably compact in the metric space ( , ), thus compact, hence closed. Space on which there exists a -metric on it is said to be -metrizable (Ščepin, 1980). The concept of -metrizability is a generalization of metrizability, in the sense that every metric induces a -metric, so every metrizable space is -metrizable. In particular, (ℝ, ) is -metrizable.
An interesting property of -metrizable spaces is that any product of -metrizable spaces is -normal. That is, any two disjoint closed domains could be separated by disjoint open sets (Ščepin, 1980). It is clear that every 4 space is -normal, so this is a weaker version of normality. Hence, we are interested in -metrizablity of the general -space and -normality of the non-normal products ofspaces.

Conclusion
We have presented some of the topological properties of the product of two H-spaces. But, these problems are still open: 1. Can the product of two H-spaces be normal yet not paracompact (equivalently, not Lindelöf)? 2. If the product of two H-spaces is not normal, is it k-normal? 3. If the product of two H-spaces is not normal, is it α-normal?

Conflict of interest
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.