Study of a Forwarding Chain in the Category of Topological Spaces between T0 and T2 with respect to One Point Compactification Operator

In the following text, we want to study the behavior of one point compactification operator in the chain Ξ := {Metrizable, Normal, T2, KC, SC, US, T1, TD, TUD, T0, Top} of subcategories of category of topological spaces, Top (where we denote the subcategory of Top, containing all topological spaces with property P , simply by P). Actually we want to know, for P ∈ Ξ andX ∈ P, the one point compactification of topological spaceX belongs to which elements of Ξ. Finally we find out that the chain {Metrizable, T2, KC, SC, US, T 1 , TD, TUD, T0, Top} is a forwarding chain with respect to one point compactification operator.


Introduction
The concept of forwarding and backwarding chains in a category with respect to a given operator has been introduced for the first time in [1] by the first author. The matter has been motivated by the following sentences in [1]: "In many problems, mathematicians search for theorems with weaker conditions or for examples with stronger conditions. In other words they work in a subcategory D of a mathematical category, namely, C, and they want to change the domain of their activity (theorem, counterexample, etc.) to another subcategory of C like K such that K ⊆ D or D ⊆ K according to their need." Most of us have the memory of a theorem and the following question of our professors: "Is the theorem valid with weaker conditions for hypothesis or stronger conditions for result?" The concept of forwarding, backwarding, or stationary chains of subcategories of a category C tries to describe this phenomenon.
In this text, Top denotes the category of topological spaces. Whenever is a topological property, we denote the subcategory of Top containing all the topological spaces with property , simply by . For example, we denote the category of all metrizable spaces by Metrizable.
We want to study the chain {Metrizable, Normal, T 2 , KC, SC, US, T 1 , T D , T UD , T 0 , Top} of subcategories of Top in the point of view of forwarding, backwarding, and stationary chains' concept with respect to one point compactification or Alexandroff compactification operator.

Remark 1.
Suppose ≤ is a partial order on . We call ⊆ (i) a chain, if for all , ∈ , we have ≤ ∨ ≤ ; (ii) cofinal, if for all ∈ , there exists ∈ such that ≤ .
In the following text, by a chain of subcategories of category C, we mean a chain under "⊆" relation (of subclasses of C). We recall that if M is a chain of subcategories of category C such that ⋃ M is closed under (multivalued) operator , then we call M where, for multivalued function , by ( ) ∈ 3 \ 2 , we mean that at least one of the values of ( ) belongs to 3 \ 2 ; (iii) a backwarding chain with respect to ; if for all ∈ M, we have ( ) ⊆ ; (iv) a full-backwarding chain with respect to ; if it is a backwarding chain with respect to and for any distinct 1 , 2 , 3 ∈ M, we have 1 ⊆ 2 ⊆ 3 ⇒ (∃ ∈ 3 \ 2 ( ) ∈ 2 \ 1 ) , (2) where, for multivalued function , by ( ) ∈ 2 \ 1 , we mean that at least one of the values of ( ) belongs to 2 \ 1 ; (v) a stationary chain with respect to if it is both forwarding and backwarding chains with respect to .
Basic properties of forwarding, backwarding, fullforwarding, full-backwarding, and stationary chains with respect to given operators have been studied in [1]. We refer the interested reader to [2] for standard concepts of the Category Theory.
We recall that by N we mean the set of all natural numbers {1, 2, . . .}; also = {0, 1, 2, . . .} is the least infinite ordinal (cardinal) number and Ω is the least infinite uncountable ordinal number. Here ZFC and GCH (generalized continuum hypothesis) are assumed (note: by GCH for infinite cardinal number , there is not any cardinal number with < < 2 , i.e., + = 2 ).
We call a collection F of subsets of a filter over if ⌀ ∉ F; for all , ∈ F we have ∩ ∈ F; for all ∈ F and ⊆ with ⊆ we have ∈ F. If F is a maximal filter over (under ⊆ relation), then we call it an ultrafilter over . If for all ∈ F, we have card( ) = card( ); then we call F a uniform ultrafilter over .
We end this section by the following two examples.

Basic Definitions in Separation Axioms
In this section we bring our basic definitions in Top.

Convention 1.
Henceforth in the topological space suppose ∞ ∉ . So (see [3,4] Regarding [5], we have T 2 ⊆ KC ⊆ SC ⊆ US ⊆ T 1 . Also by [6] we have In this section, we want to study the operator on the above chain. However, it has been proved in [1, Lemma 3.1 and Corollary 3.2] that the chain T 1 ⊆ T D ⊆ T UD ⊆ T 0 is stationary with respect to the operator ; therefore, the main interest is on Metrizable ⊆ Normal ⊆ T 2 ⊆ KC ⊆ SC ⊆ US ⊆ T 1 .

Note 1. A topological space is KC if and only if { ⊆ :
is an open subset of } ∪ { ∪ {∞} : ⊆ and \ is a compact subset of } is a topological basis on . (3) ( ) is T 2 if and only if is T 2 and locally compact [4]; thus ( ) is T 2 if and only if it is normal.
(5) If ( ) is KC, then is KC too (hint: if is a compact subset of , then is a compact subset of ( ) by (2). If ( ) is KC, then is a closed subset of ( ), and again by (2), is a closed subset of , so is KC).
(6) A T 2 space is a -space if it is either first countable or locally compact so every metrizable space is -space [3,7]. For topological spaces , , by ⊔ , we mean topological disjoint union of and .  (iv) Consider ∞ ∈ . In this case, 1 := ∩ 1 is an open subset of 1 by Remark 3(4). Using the compactness of 1 , 1 \ 1 is a closed compact subset of 1 . Also is an open subset of ( 2 ) containing ∞; thus 2 \ 2 is a closed compact subset of 2 . Since 1 \ 1 and 2 \ 2 are two closed compact subsets of Hence is an open subset of ( 1 ⊔ 2 ).

Lemma 5. If is a closed subset of , then ( ) is an embedding of ( ).
Proof. If is compact, then ( ) = and by Remark 3(4) we are done. If is not compact, \ is an open subset of and ( ); thus ∪ {∞} is a closed compact subset of ( ). Suppose ⊆ ∪ {∞}; we prove that is a closed subset of * := ∪{∞} as a subspace of ( ) if and only if is a closed subset of ( ) = ∪ {∞} as one point compactification of . However, we mention that ∪ {∞} in both topologies is an embedding of by Remark 3(4).
First, suppose is a closed subset of * . Using the following two cases, is a closed subset of ( ) too.
(i) Consider ∞ ∈ . In this case, := * \ = \ is an open subset of ; therefore it is an open subset of ( ), so = ( ) \ is a closed subset of ( ).
(ii) Consider ∞ ∉ . In this case, is a closed subset of ( ) since it is a closed subset of * and * is closed in ( ). Therefore, Conversely, suppose is a closed subset of ( ). Using the following two cases, is a closed subset of * too. (iv) Consider ∞ ∉ . In this case, is a closed compact subset of ( ) with ∞ ∉ ; thus is a closed compact subset of . Hence, is a closed compact subset of , and = ( ) \ is an open subset of ( ). Therefore, ∩ * = * \ is an open subset of * , so is a closed subset of * .
We have the following.
(1) has a formal proof, so we deal with (2). If ∈ C and is a closed subspace of , then ∈ C. Suppose , ∈ C; , are closed subspaces of with ∩ = { } and ∪ = . We prove ∈ C. (i) Consider C = Metrizable. If , are metrizable subspaces of , then there exist metrics 1 , 2 , respectively, on , such that 1 , 2 ≤ 1, the metric topology induced from 1 on is subspace topology on induced from , and the metric topology induced from 2 on is subspace topology on induced from . Define : × → [0, +∞) with Then the metric topology induced from on coincides with 's original topology.
(ii) Consider C = T 2 . Suppose , are Hausdorff subspaces of and , ∈ are two distinct points of . Consider the following cases:  1 , are disjoint open subsets of with ∈ 1 and ∈ .
Using the above cases, is Hausdorff.
(iii) Consider C = Normal. If , are normal subspaces of , then , are Hausdorff and, using the case "C = T 2 ", is Hausdorff. Now suppose , are disjoint closed subsets of ; also we may suppose ∉ . (iv) Consider C = KC. Suppose , are KC and is a compact subset of . Since , are closed, ∩ , ∩ are compact too. Since ∩ is a compact subset of and is KC, ∩ is a closed subset of . Since ∩ is a closed subset of and is a closed subset of , ∩ is closed subset of . Similarly, ∩ is a closed subset of . Thus = ( ∩ ) ∪ ( ∩ ) is a closed subset of and is KC.
(v) Consider C = SC. Suppose , are SC and ( : ∈ ) is a sequence in converging to . Using the following cases, { : ∈ } ∪ { } is a closed subset of .
, { } is a closed subset of (resp. ) since (resp. ) is SC and in particular We may suppose ∈ . Since is T 1 , { } is a closed subset of . Since is a closed subset of and { } is a closed subset of , { } is a closed subset of .
(viii) Use similar methods for the rest of the cases of C. Proof. Let be a noncompact SC space. Suppose ( : ∈ ) is a sequence in ( ) = ∪ {∞} converging to , ∈ ( ). We have the following cases.
(i) Consider , ∈ . In this case, is an open neighborhood of , in ( ); hence there exists ∈ such that ∈ for all ≥ . Therefore, ( : ≥ ) is a converging sequence in to , . Since is SC, is US and = .
(ii) Consider ∈ , = ∞. In this case, there exists ∈ such that ∈ for all ≥ . Therefore, ( : ≥ ) is a converging sequence in to . Thus ∉ for all ≥ and by converging ( : ∈ ) to . So this case does not occur.
Using the above cases, we have = , and ( ) is US.

The Main Table
See Figure 1; then we have Table 1 which we prove in this Section and where: The mark "√" indicates that in the corresponding case, there exists ∈ such that ( ) ∈ , and the mark "-" indicates that in the corresponding case for all ∈ we have ( ) ∉ . Let By Remark 3 (7) in Table 1, the mark "-" for cases in which " ∈ E, ∈ F" or " ∈ F, ∈ E" is evident. However, it has been proved in [1, Lemma 3.1 and Corollary 3.2] that the chain T 1 ⊆ T D ⊆ T UD ⊆ T 0 is stationary with respect to the operator , so corresponding marks of the cases in which , ∈ F are obtained. Thus it remains to discuss cases in which , ∈ E.
Since the subspace of a metrizable (resp. T 2 , SC, and US) space is metrizable (resp. T 2 , SC, and US) using Remark 3 (4) and (5), if ( ) is, respectively, metrizable T 2 , KC, SC, or US, then is too. Hence we obtain "-" for the following cases too (choose and from the same rows of Table 2). Figure 1).
Consider as the set of all rational numbers as a subspace of Euclidean space R. Since is not locally compact, by Remark 3(3), ( ) is not Hausdorff. Suppose is a compact subset of ( )); in order to show that ( ) is KC, we show is a closed subset of ( ). We have the following two cases. Case 1. If ∞ ∉ , then is a compact subset of ; since is a metric space, is a closed subset of too. Therefore, is an open subset of ( ). Hence, is a closed subset of ( ). is an open subset of ( ). Since ∉ , ⊆ ⋃{ : ≥ 0}. Using the compactness of , there exists ≥ 1 such that Metrizable C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 C 11 C 9 := T UD spaces   Table 2 Reason of omitting this case If ( ) is metrizable, then is metrizable too If ( ) is SC, then is SC too If ( ) is US, then is US too is a contradiction. Thus \ is an open subset of , and = ( ) \ ( \ ) is a closed subset of ( ).
Using the above three cases, { : ∈ } ∪ { } is a closed subset of ( ) and we are done.
Second Row. Here we have = 2 \ 1 and the following cases for .
(iv) Consider On the other hand, using the definition of one point compactification, any subset of ( ) containing ∞ is a compact subset of ( ). Therefore, ( ) \ { } is a compact subset of ( ), but it is not a closed subset of ( ); thus ( ) is not KC. We claim that ( ) is SC. Suppose ( : ∈ ) is a sequence in ( ) converging to . We have the following cases. it is an open subset of ( ). Thus { : ∈ }∪{ } is a closed subset of ( ).
Since 0 ∈ \ {0} and by the above two cases, there is not any sequence in \ {0} converging to 0; is not metrizable. Thus ∈ 2 \ 1 . Now pay attention to the following claims. Third Row. Here we have = 3 \ 2 and the following cases for .
(v) Consider = 6 \ 5 . Consider as disjoint union of 1 and 2 , where we have the following.
Fourth Row. Here we have = 4 \ 3 and the following cases for .
(ii) Consider = 5 \ 4 . Consider uncountable set with countable complement topology { ⊆ : = ⌀∨( \ is countable)} [4, counterexamples 20 and 21]. Since every two nonempty open subsets of have nonempty intersection, is not Hausdorff. It is clear that is T 1 . Moreover, is a compact subset of if and only if is finite. Therefore, every compact subset of is closed and is KC. So ∈ 4 \ 3 . Now suppose is an uncountable subset of with uncountable complement. So is not closed. For all compact subset of , the set ∩ is finite and closed. Therefore, is not a -space. Using Remark 3(2), ( ) is not KC. Using Remark 3(1), ( ) is US; we claim that ( ) is SC. Suppose ( : ∈ ) is a sequence in ( ), converging to ∈ ( ). We have the following cases.   Table 1 regarding case " = 2 \ 1 , = 6 \ 5 ").

Some Observations in
(i) The collection {T 2 , KC, SC, T 1 } is a full-forwarding chain with respect to . In other words, Table 3 is valid.
In Table 3, the mark "√" indicates that in the corresponding case there exists ∈ such that ( ) ∈ , and the mark "-" indicates that in the corresponding case for all ∈ we have ( ) ∉ .
(ii) The collection {Metrizable, T 2 , KC, SC, US, T 1 , T D , T UD , T 0 , Top} is a forwarding chain with respect to . The collection T 1 , T D , T UD , T 0 , Top is a stationary chain with respect to .