View of AS and AS_2-Paracompactness in Topological Spaces

𝐀𝐒 and 𝐀𝐒

-Paracompactness in Topological Spaces

1S.Umamaheswari, ²M.Saraswathi

1,2Assistant Professor of Mathematics K.K.C, P.Velur,Namakkal, Tamil Nadu, India

1[email protected] , ²[email protected]

Abstract—The purpose of this paper is to introduce the two new concepts of 𝑨𝑺-Paracompact spaces and 𝑨𝑺_𝟐-Paracompact spaces. Also we have proved that every 𝑨𝑺-Paracompactness and 𝑨𝑺_𝟐-Paracompactness has a topological property. Furthermore, we have introduced 𝐀𝐒-normal and its properties.

Keywords—Angelic spaces, seperable,𝑺-Paracompact,𝑺_𝟐–Paracompact, 𝑺-Normal, 𝑨𝑺- Paracompact, 𝑨𝑺_𝟐–Paracompact, 𝑨𝑺-normal.

MSC (2010): 46A50, 54C10, 54D15, 54D20.

I. INTRODUCTION

In this paper, we introduce two new properties in topological spaces which are 𝐴𝑆- paracompactness and 𝐴𝑆₂-paracompactness. And also introduced the concept of 𝐴𝑆-normal.

Diedonne.Jintroduced by [4] Paracompact space. Arhangelskii defined by𝐶-paracompact and 𝐶₂-paracompact.𝐶-paracompact and 𝐶₂-paracompact were studied in [15]. Arhangelskii introduced on 2012 a new weaker version of normality issaid to 𝐶-normality [2]. In 2019, 𝑆- paracompactness and 𝑆₂-paracompactness were studied in [13] and Lutfi Kalantan [8]

introduced 𝑆-normality. Fermlin. D.H [3] introduced the angelic space. The introduction of an angelic space was a step in the study of compactness. In this paper we are using angelic spaces in 𝑆-paracompact, 𝑆₂-paracompact and 𝑆-normal. Angelic paracompact space represented by 𝒜𝒫𝒮 and angelic paracompact denoted by 𝒜𝒫.

II. PRELIMINARIES Definition 2.1 :[3]

A 𝑇𝑆 is said to bean angelic, if for each relatively countably compact subset 𝐾 of 𝐹 the following hold: (a) 𝐾 is relatively compact (b) If 𝑘 ∈ 𝐾, then ∃a sequence in 𝐾 that converges to 𝑘.

Definition 2.2:[12]

A 𝑇𝑆 is termed asseperable if it has a countable dense subset.

Definition 2.3:[13]

Let 𝐹 be a TS and 𝐾 be an angelic seperable subspace of 𝐹. If there is a bijection mapping 𝑟: 𝐹 → 𝐺, 𝐺 is an paracompact(Hausdorff paracompact) space and the restriction 𝑟 _𝑘 : 𝐾 → 𝑟(𝐾) is a homeomorphism, then 𝐹 is said to be a𝑺-paracompact(𝑺_𝟐-paracompact) space.

Definition 2.4:[11]

ATS𝐹 is termed asa P-space if 𝐹 is 𝑇₁ and each𝐺_𝛿-set is open.

Definition 2.5:[6]

A TS𝐹 is said to be locally seperable, if each element has a seperable open neighbourhood.

Definition 2.6:[14]

A space 𝐹 is of countable tightness if for every subset 𝐾 of 𝐹 and every𝑓 ∈ 𝐹 with 𝑓 ∈ 𝐾∃ a countable subset 𝐿 ⊆ 𝐹 ∋ 𝑙 ∈ 𝐿

III. 𝑨𝑺-PARACOMPACTNESS AND 𝑨𝑺_𝟐-PARACOMPACTNESS

Definition 3.1:

Let 𝑋 be an angelic space and 𝐴 be an angelic seperable subspace of 𝑋. If ∃ a bijection mapping 𝑓: 𝑋 → 𝑌,𝑌 is an 𝒜𝒫𝒮 and the restriction 𝑔 _𝒜: 𝒜 → 𝑔(𝒜) is a homeomorphism, then 𝒫 is said to be an 𝑨𝑺-paracompact space.

Definition 3.2:

Let 𝒫 be an angelic space and 𝒜 be an angelic seperable subspace of 𝒫. If there is a bijection mapping 𝑔: 𝒫 → 𝒬, 𝒬 is Hausdorff 𝒜𝒫𝒮 and the restriction 𝑔 _𝒜: 𝒜 → 𝑔(𝒜) is a homeomorphism, then 𝒫 is said to be an 𝑨𝑺_𝟐-paracompact space.

Theorem 3.3:

Let𝐹 be an angelic separable but not 𝑇₂, then 𝐹 cannot be an 𝐴𝑆₂-paracompact.

Proof.

Suppose𝐹 is any separable non-𝑇₂ space. Let𝐹be an𝐴𝑆₂-paracompact. Then

∃ a𝑇₂𝒜𝒫𝒮𝐺 and a bijective function 𝑟: 𝐹 → 𝐺 ∋ 𝑟 _𝐾: 𝐾 → 𝑟(𝐾) is a homeomorphism for all angelic separable subspaces𝐾 ⊆ 𝐹. Since 𝐹 is separable, then 𝑟: 𝐹 → 𝐺 is a homeomorphism.

But 𝐺 is Hausdorff, then 𝐹 is Hausdorff which is a contradiction. Hence 𝐹 cannot be 𝐴𝑆₂- paracompact.

Example 3.4:

Let ℝ, 𝒞ℱ be the finite complement topology defined on the real numbers. Since (ℝ, 𝒞ℱ) is an 𝒜𝒫 being an angelic compact, then the identity function 𝑖𝑑: ℝ, 𝒞ℱ → (ℝ, 𝒞ℱ) shows that it is an𝐴𝑆-paracompact but not an𝐴𝑆₂-paracompact being angelic separable but not 𝑇₂ space.

Example 3.5.

Ω₁ is an 𝐴𝑆₂-paracompact space which is not an angelic paracompact. Any angelic separable subspace of Ω₁ is countable. Suppose 𝐾 ⊂Ω₁ is uncountable which implies that 𝐾 is unbounded in Ω₁. If 𝑁 is any countable subset of 𝐾, then ∃ 𝛼 < Ω₁ ∋ 𝛼 = 𝑆𝑢𝑝 𝑁. Thus,

∃ 𝜂 ∈ 𝐾 ∋ 𝛼 < 𝜂 . The set ( 𝛼, 𝜂 ∩ 𝐾) is a nonempty open subset of 𝐾 with 𝛼, 𝜂 ∩ 𝐾 ∩ 𝑁 = ∅. Thus, 𝐾 cannot be an angelic separable implying that any angelic separable subspace of Ω₁ is countable. Since Ω₁ is 𝑇₂ locally angelic compact, then ∃ a one to one continuous function, say 𝑟, onto a Hausdorff compact space 𝐺. Let 𝐾 ⊂Ω₁ be any angelic separable subspace. Since the closure of any countable set is an angelic compact in Ω₁, then we get that 𝑟 _𝐾: 𝐾 → 𝑟(𝐾) is a homeomorphism implying 𝑟 _𝐾: 𝐾 → 𝑟(𝐾) is a homeomorphism.

Theorem 3.6:

If 𝐹 is an angelic separable but not an angelic paracompact, then 𝐹 cannot be an 𝐴𝑆- paracompact.

Proof:

Suppose𝐹 is any angelic separable non-𝒜𝒫𝒮. Let𝐹 be an𝐴𝑆-paracompact. Then

∃ an𝒜𝒫𝒮𝐺 and a bijective function 𝑟: 𝐹 → 𝐺 ∋ 𝑟 _𝐾: 𝐾 → 𝑟(𝐾) is a homeomorphism for everyangelic separable subspaces𝐾 ⊆ 𝐹. Since 𝐹 is an angelic separable, then 𝑟: 𝐹 → 𝐺 is a homeomorphism. But 𝐺 is an 𝒜𝒫, then 𝐹 is 𝒜𝒫 which is a contradiction. Hence 𝐹 cannot be an 𝐴𝑆 -paracompact.

Theorem 3.7:

Every 𝑇₂ countably compact angelic separable 𝐴𝑆 -paracompact space isan angelic compact.

Proof:

Suppose𝐹is any 𝑇₂ countably angelic compact angelic separable 𝐴𝑆 -paracompact space. Then 𝐹 is an 𝒜𝒫 because the witness function of 𝐴𝑆 -paracompactness is a homeomorphism. Since any countably angelic compact 𝑇₂𝒜𝒫𝒮 is an angelic compact.

Hence𝐹 is an angelic compact.

Theorem 3.8:

If 𝐹 is an 𝐴𝑆 -paracompact (𝐴 𝑆₂-paracompact) space and Fr´echet such that 𝑟 ∶ 𝐹 → 𝐺 is a witness of 𝐴𝑆 -paracompactness (𝐴𝑆 ₂-paracompactness) of 𝐹, then 𝑟 is continuous.

Proof:

Suppose𝐹 is an𝐴𝑆 -paracompact and Frechet. Let 𝑟 ∶ 𝐹 → 𝐺 witnesses 𝐴𝑆 - paracompactness of 𝐹. Let 𝐾 ⊆ 𝐹 and pick 𝑔 ∈ 𝑟 (𝐾 ) implying that ∃ 𝑎 a unique 𝑓 ∈ 𝐹 ∋ 𝑟 (𝑓 ) = 𝑔 and 𝑓 ∈ 𝐾. Since𝐹 is Frechet, then ∃a sequence 𝑎_𝑛 ⊆ 𝐾 ∋ 𝑎_𝑛 →𝑓. The subspace 𝐿 = {𝑓 , 𝑎_𝑛 ∶ 𝑛 ∈ 𝑁} of 𝐹 is an angelic separable by being countable, thus 𝑟 _𝐿: 𝐿 → 𝑟 (𝐿 ) is a homeomorphism. Now, let 𝑃 ⊆ 𝐺 be any open neighborhood of 𝑔. Then 𝑃 ∩ 𝑟 (𝐿 ) is open in the subspace 𝑟 𝐿 ⊆ 𝑔. Since 𝑟 ({𝑎_𝑛 ∶ 𝑛 ∈ 𝑁}) ⊆ 𝑟 (𝐿 ) ∩ 𝑟 (𝐾) and 𝑃 ∩ 𝑟 𝐿 ≠ ∅ , 𝑃 ∩ 𝑟 𝐿 ≠ ∅ . Hence 𝑔 ∈ 𝑟 (𝐾) , thus 𝑟 (𝐾 ) ⊆ 𝑟 (𝐴 ) . Therefore, 𝑟 is continuous.

Theorem 3.9:

𝐴𝑆 -paracompactness (𝐴 𝑆₂-paracompactness) is a topological property.

Proof:

Let 𝐹 be an 𝐴𝑆 -paracompact space and 𝐻 be any TS∋𝐹 is homeomorphic to 𝐻. Let 𝑟 be the function witnessing 𝐴𝑆 -paracompactness of 𝐹 onto an 𝒜𝒫𝒮𝐺 and 𝑡 ∶ 𝐹 → 𝐺 be a homeomorphism. Then 𝑟 ∘ 𝑡 ⁻¹ ∶ 𝐻 → 𝐺 will be the witness of 𝐴𝑆 -paracompactness of 𝐻. Hence 𝐴𝑆 -paracompactness (𝐴 𝑆₂-paracompactness) is a topological property.

Theorem 3.10:

The Cartesian product of two 𝐴 𝑆₂-paracompact spaces is 𝐴 𝑆₂-paracompact in case that at least one of them is countably compact and Fr´echet.

Proof:

Let 𝐹 and 𝐻 be 𝐴𝑆 ₂-paracompact ∋ 𝐹 is countably compact and Frechet. Let𝐺 and 𝑟 ∶ 𝐹 → 𝐺 be witnesses of 𝐴 𝑆₂-paracompactness of 𝐹. Then 𝑟 is continuous by Theorem 3.8 implying that𝐺 is countably angelic compact. Hence, 𝐺 is compact. Let 𝐺′ and 𝑟 ′∶ 𝐻 → 𝐺′ be witnesses of 𝐴 𝑆₂-paracompactness of 𝐻. Consider the function 𝑡 ∶= 𝑟 × 𝑟 ′: 𝐹 × 𝐻 → 𝐺 × 𝐺 ′ . Observe that 𝐺 × 𝐺′ is 𝑇₂angelic paracompact. Now, let 𝑁 be any angelic separable subspace of 𝐹 × 𝐻. Therefore, 𝑝₁(𝑁) ⊆ 𝐹 and 𝑝₂(𝑁) ⊆ 𝐻 are both separable subspaces of 𝐹 and𝐻 respectively being continuous images of an angelic separable subspace 𝑁 ⊆ 𝐹 × 𝐻. Then countable product of an angelic separable spaces is angelic separable, 𝑝₁(𝑁) × 𝑝₂(𝑁) is an angelic separable in 𝐹 × 𝐻. Thus, as 𝑁 ⊆ 𝑝₁(𝑁) × 𝑝₂(𝑁), we get that 𝑡 _𝐿: 𝐿 → 𝑡 (𝐿 ) is a homeomorphism. Hence the Cartesian product of two 𝐴 𝑆₂-paracompact spaces is 𝐴 𝑆₂-paracompact in case that at least one of them is countably compact and Fr´echet.

Theorem 3.11:

𝐴𝑆 -paracompactness (𝐴 𝑆₂-paracompactness) is an additive property.

Proof:

Suppose {𝐹_𝛼 ∶ 𝛼 ∈ 𝛬 } is a family of 𝐴𝑆 ₂ -paracompact spaces. Hence, for every𝛼 ∈ 𝛬 ∃an𝒜𝒫𝒮 𝐺_𝛼 and a bijection 𝑟_𝛼 ∶ 𝐹_𝛼 → 𝐺_𝛼 ∋𝑟_{𝛼 𝐾𝛼} ∶ 𝐾_𝛼 → 𝑟_𝛼(𝐾_𝛼) is a homeomorphism for everyangelic separable subspace 𝐾𝛼 of 𝐹𝛼. Since angelic paracompactness is an additive property,⊕_𝛼∈𝛬 𝐺_𝛼 is an angelic paracompact. Define 𝑟 ∶ ⊕_𝛼∈𝛬 𝐹 → ⊕_𝛼∈𝛬 𝐺_𝛼 as follows: for each 𝑓 ∈⊕_𝛼∈𝛬 𝐹_𝛼, there exists a unique 𝛾 ∈ 𝛬 such that 𝑓 ∈ 𝐹_𝛾 then 𝑟 (𝑓 ) = 𝑟_𝛾(𝑓 ). Let 𝐾 be any angelic separable subspace of

⊕_𝛼∈𝛬 𝐹_𝛼. Write 𝐾 = ∪ 𝛼 ∈ 𝛬^∗ (𝐾 ∩ 𝐹_𝛼) where 𝛬^∗ = {𝛼 ∈ 𝛬 ∶ 𝐾 ∩ 𝐹_𝛼 ,∅ }. Since 𝐾 is an angelic separable, then 𝛬^∗ is countable and 𝐾 ∩ 𝐹_𝛼is an angelic separable in 𝐹_𝛼 for all 𝛼 ∈ 𝛬^∗. Therefore, 𝑟𝛼 𝐾∩𝑋_𝛼 ∶ 𝐾 ∩ 𝐹𝛼 → 𝑟𝛼(𝐾 ∩ 𝐹𝛼) is a homeomorphism for every𝛼 ∈ 𝛬^∗ implying that 𝑟 _𝐾: 𝐾 → 𝑟 (𝐾) is a homeomorphism.

Theorem3.12:

Every second countable 𝐴 𝑆₂-paracompact space is metrizable.

Proof:

Suppose(𝐹 , 𝜏 ) is an 𝐴 𝑆₂-paracompact second countable space which yields that 𝐹 is an angelic separable 𝐴𝑆 ₂-paracompact. Then 𝐹 is 𝑇₄ implying that 𝐹 is regular. Since any second countable 𝑇₃ space is metrizable, we get that 𝐹 is metrizable.

Theorem 3.13:

Every P-space is 𝐴𝑆 ₂-paracompact.

Proof:

Suppose(𝐹 , 𝜏 ) is a P-space. If 𝐹 is countable, then it is discrete implying that 𝐹 is an𝐴𝑆 ₂-paracompact. Assume 𝐹 is uncountable. Let 𝐾 ⊆ 𝐹 be an arbitrary uncountable subset of 𝐹 and let 𝑁 ⊂ 𝐾 be any countable subset of 𝐾. Then 𝑁 is closed set in 𝐾 and 𝐾 \𝑁 is a non-empty open set in 𝐾 with (𝐾 \𝑁) ∩ 𝑁 = ∅. Hence, 𝑁 cannot be dense in 𝐾 implying that 𝐾 cannot be separable. Thus, any angelic separable subspace of 𝐹 must be countable and hence any angelic separable subspace of 𝐹 is discrete. Take the identity map 𝑖𝑑 ∶ (𝐹 , 𝜏 ) → (𝐹 , 𝑁). Then 𝑖𝑑 _𝐾: 𝐾 → 𝑟 (𝐾) is a homeomorphism for everyangelic separable subspaces 𝐾. Hence, (𝐹 , 𝜏 ) is an𝐴𝑆 ₂-paracompact.

Example 3.14:

Let(ℝ,𝐶𝐶 ), where 𝐶𝐶 is the countable complement topology defined on ℝ. Since (ℝ,𝐶𝐶 ) is P-space, then by Theorem 3.13, (ℝ,𝐶𝐶 ) is 𝐴𝑆 ₂-paracompact. Note function witnessing the 𝐴𝑆 ₂-paracompactness here is the identity taken from 𝐶𝐶 to the discrete topology defined on ℝ, write (ℝ,𝐷). However, 𝑖𝑑 ∶ (ℝ,𝐶𝐶 ) → (ℝ,𝐷) is not continuous.

IV. 𝑨𝑺 -NORMAL AND ITS PROPERTIES

Definition 4.1:

A 𝑇𝑆𝐹 is termed as an𝐴𝑆 -normal if ∃ a normal space 𝐺 and a bijective function 𝑟 ∶ 𝐹 → 𝐺 ∋ the restriction 𝑟 _𝐾 ∶ 𝐾 → 𝑟 (𝐾) is a homeomorphism for everyangelic separable subspace 𝐾 ⊆ 𝐹.

Example 4.2:

Consider the left ray topology defined on ℝ, (ℝ,𝐿 ) such that 𝐿 = ∅ ,ℝ ∪ { −∞,𝑎 : 𝑎 ∈ ℝ}. It is an example of 𝐴𝑆 -normal space by being a normal space which is not 𝐴𝑆 ₂-paracompact space since it is separable and not 𝑇₂ space. In fact, it is not even an 𝐴𝑆 -paracompact because it is separable not an𝒜𝒫𝒮 .

Theorem 4.3:

𝐴𝑆 -normality has a topological property.

Proof:

Suppose𝐹 is an 𝐴𝑆 -normal space and 𝐹 ≅ 𝐻 and𝐺 is a normal space and 𝑟 ∶ 𝐹 → 𝐺 be a bijective function ∋𝑟 𝑀 ∶ 𝑀 → 𝑟 (𝑀) is a homeomorphism for everyangelic separable subspace 𝑀 of 𝐹. Let 𝑡 ∶ 𝐻 → 𝐹 be a homeomorphism. Hence 𝐺 and𝑟 ∘ 𝑡 ∶ 𝐻 → 𝐺has the topological property.

Theorem 4.4:

𝐴𝑆 -normality is an additive property.

Proof:

Suppose𝐹_𝛼 be an 𝐴𝑆 -normal space for every𝛼 ∈ 𝛬. Then their sum ⊕_𝛼∈𝛬 𝐹_𝛼is 𝐴𝑆 -normal. For every𝛼 ∈ 𝛬, pick a normal space 𝐺𝛼 and a bijective function 𝑟𝛼 ∶ 𝐹𝛼 → 𝐺_𝛼 ∋𝑟_𝛼

𝑀𝛼 ∶ 𝑀_𝛼 → 𝑟_𝛼(𝑀_𝛼) is a homeomorphism for each angelic separable subspace 𝑀_𝛼 of 𝐹_𝛼. Since 𝐺_𝛼 is normal for every𝛼 ∈ 𝛬, the sum ⊕_𝛼∈𝛬 𝐺_𝛼 is normal. Consider the function sum, ⊕_𝛼∈𝛬 𝑟_𝛼:⊕_𝛼∈𝛬 𝐹_𝛼 →⊕_𝛼∈𝛬 𝐺_𝛼 defined by ⊕_𝛼∈𝛬 𝑟_𝛼(𝑓 ) = 𝑟 _𝛽 (𝑓 ) if 𝑓 ∈ 𝐹_𝛽, 𝛽 ∈ 𝛬. Now, a subspace 𝑀 ⊆⊕_𝛼∈𝛬 𝐹_𝛼is an angelic separable if and only if the set 𝛬₀ = {𝛼 ∈ 𝛬 ∶ 𝑀 ∩ 𝐹_𝛼 ≠ ∅ } is countable and 𝑀 ∩ 𝐹_𝛼 is angelic separable in 𝐹_𝛼 for each 𝛼 ∈ 𝛬₀ . If 𝑀 ⊆⊕_𝛼∈𝛬 𝐹_𝛼 is an angelic separable, then ⊕_𝛼∈𝛬 𝑟_{𝛼 𝑀} is a homeomorphism because 𝑟_𝛼 𝑀 ∩ 𝐹_𝛼 is a homeomorphism for every𝛼 ∈ 𝛬₀.

Theorem 4.5:

If 𝐹 is an𝐴𝑆 -normal and of countable tightness and 𝑟 ∶ 𝐹 → 𝐺 witnesses the 𝐴𝑆 - normality of 𝐹. Then 𝑟 is continuous.

Proof:

Suppose that 𝐹 is an 𝐴𝑆 -normal and of countable tightness. Let 𝑟 ∶ 𝐹 → 𝐺 be any witness of the 𝐴𝑆 -normality of 𝐹 . Let 𝐾 ⊆ 𝐹 be arbitrary. We have 𝑟 (𝐾 ) = 𝑟 ( _{𝐿∈ 𝐾 ≤Ω0}𝐿 ) = _{𝐿∈ 𝐾 ≤Ω0}𝑟 𝐿 ⊆ _{𝐿∈ 𝐾 ≤Ω} 𝑟 (𝐿 )

0 ⊆ 𝑟 (𝐾) . Therefore, 𝑟 is continuous.

Remark:

Any first countable space is Fr𝑒 chet, any Fr𝑒 chet space is sequential and any sequential space is of countable tightness in [6] we conclude the following corollary.

Corollary 4.6:

(1) If 𝐹 is 𝐴𝑆 -normal first countable and 𝑟 ∶ 𝐹 → 𝐺 witnesses the 𝐴𝑆 -normality of 𝐹. Then 𝑟 is continuous.

(2) If 𝐹 is 𝐴𝑆 -normal Fr𝑒 chet and 𝑟 ∶ 𝐹 → 𝐺 witnesses the 𝐴𝑆 -normality of 𝐹. Thenr𝑓 is continuous.

(3) If 𝐹 is 𝐴𝑆 -normal sequential and 𝑟 ∶ 𝐹 → 𝐺 witnesses the 𝐴𝑆 -normality of 𝐹. Then 𝑟 is continuous.

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