### 𝐀𝐒 and 𝐀𝐒

_{𝟐}

**-Paracompactness in Topological Spaces **

*1**S.Umamaheswari, *^{2}*M.Saraswathi *

*1,2**Assistant 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 𝐴𝑆_{2}-paracompactness. And also introduced the concept of 𝐴𝑆-normal.

Diedonne.Jintroduced by [4] Paracompact space. Arhangelskii defined by𝐶-paracompact and
𝐶_{2}-paracompact.𝐶-paracompact and 𝐶_{2}-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 𝑆_{2}-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, 𝑆_{2}-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 𝑇_{1} 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 𝑇_{2}, then 𝐹 cannot be an 𝐴𝑆_{2}-paracompact.

*Proof. *

Suppose𝐹 is any separable non-𝑇_{2} space. Let𝐹be an𝐴𝑆_{2}-paracompact. Then

∃ a𝑇_{2}𝒜𝒫𝒮𝐺 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 𝐴𝑆_{2}-
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 an𝐴𝑆_{2}-paracompact being angelic separable but
not 𝑇_{2 }space.

*Example 3.5. *

Ω_{1} is an 𝐴𝑆_{2}-paracompact space which is not an angelic paracompact. Any angelic
separable subspace of Ω_{1} is countable. Suppose 𝐾 ⊂Ω_{1} is uncountable which implies that 𝐾
is unbounded in Ω_{1}. If 𝑁 is any countable subset of 𝐾, then ∃ 𝛼 < Ω_{1} ∋ 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 Ω_{1} is countable. Since Ω_{1} is 𝑇_{2} locally angelic compact, then ∃ a one to one continuous
function, say 𝑟, onto a Hausdorff compact space 𝐺. Let 𝐾 ⊂Ω_{1} be any angelic separable
subspace. Since the closure of any countable set is an angelic compact in Ω_{1}, 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 𝑇* _{2}* countably compact angelic separable 𝐴𝑆 -paracompact space isan angelic
compact.

*Proof: *

Suppose𝐹is any 𝑇* _{2}* 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 𝑇

*𝒜𝒫𝒮 is an angelic compact.*

_{2}Hence𝐹 is an angelic compact.

*Theorem 3.8: *

If 𝐹 is an 𝐴𝑆 -paracompact (𝐴 𝑆* _{2}*-paracompact) space and Fr´echet such that
𝑟 ∶ 𝐹 → 𝐺 is a witness of 𝐴𝑆 -paracompactness (𝐴𝑆

*-paracompactness) of 𝐹, then 𝑟 is continuous.*

_{2}*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 𝑆* _{2}*-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 𝑆* _{2}*-paracompactness) is a topological property.

*Theorem 3.10: *

The Cartesian product of two 𝐴 𝑆* _{2}*-paracompact spaces is 𝐴 𝑆

*-paracompact in case that at least one of them is countably compact and Fr´echet.*

_{2}*Proof: *

Let 𝐹 and 𝐻 be 𝐴𝑆 * _{2}*-paracompact ∋ 𝐹 is countably compact and Frechet. Let𝐺 and
𝑟 ∶ 𝐹 → 𝐺 be witnesses of 𝐴 𝑆

*-paracompactness of 𝐹. Then 𝑟 is continuous by Theorem 3.8 implying that𝐺 is countably angelic compact. Hence, 𝐺 is compact. Let 𝐺*

_{2}*′ and*𝑟 ′∶ 𝐻 → 𝐺

*′ be witnesses of*𝐴 𝑆

*-paracompactness of 𝐻. Consider the function 𝑡 ∶= 𝑟 × 𝑟 ′: 𝐹 × 𝐻 → 𝐺 × 𝐺 ′ . Observe that 𝐺 × 𝐺*

_{2}*′ is 𝑇*

*angelic paracompact. Now, let 𝑁 be any angelic separable subspace of 𝐹 × 𝐻. Therefore, 𝑝*

_{2}*(𝑁) ⊆ 𝐹 and 𝑝*

_{1}*(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, 𝑝*

_{2}*(𝑁) × 𝑝*

_{1}*(𝑁) is an angelic separable in 𝐹 × 𝐻. Thus, as 𝑁 ⊆ 𝑝*

_{2}*(𝑁) × 𝑝*

_{1}*(𝑁), we get that 𝑡*

_{2}_{𝐿}: 𝐿 → 𝑡 (𝐿 ) is a homeomorphism. Hence the Cartesian product of two 𝐴 𝑆

*-paracompact spaces is 𝐴 𝑆*

_{2}*-paracompact in case that at least one of them is countably compact and Fr´echet.*

_{2}*Theorem 3.11: *

𝐴𝑆 -paracompactness (𝐴 𝑆* _{2}*-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}

_{𝐾}

_{𝛼}∶ 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_{𝛼}is 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 𝑆* _{2}*-paracompact space is metrizable.

*Proof: *

Suppose(𝐹 , 𝜏 ) is an 𝐴 𝑆* _{2}*-paracompact second countable space which yields that 𝐹 is
an angelic separable 𝐴𝑆

*-paracompact. Then 𝐹 is 𝑇*

_{2}*implying that 𝐹 is regular. Since any second countable 𝑇*

_{4}*space is metrizable, we get that 𝐹 is metrizable.*

_{3}*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

*-paracompact.*

_{2}*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

*-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.*

_{2}**IV. ** 𝑨𝑺 **-N****ORMAL 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 𝑇

*space. In fact, it is not even an 𝐴𝑆 -paracompact because it is separable not an𝒜𝒫𝒮 .*

_{2}*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_{𝛼}is
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_{𝛼}is an angelic separable if and only if the
set 𝛬* _{0}* = {𝛼 ∈ 𝛬 ∶ 𝑀 ∩ 𝐹

_{𝛼}≠ ∅ } is countable and 𝑀 ∩ 𝐹

_{𝛼}is angelic separable in 𝐹

_{𝛼}for each 𝛼 ∈ 𝛬

*. If 𝑀 ⊆⊕*

_{0}_{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 ∈ 𝛬

*.*

_{0}*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}_{ ≤}_{Ω}* _{0}*𝐿 ) =

_{𝐿}

_{∈ }

_{𝐾}

_{ ≤Ω}

*𝑟 𝐿 ⊆*

_{0}_{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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