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๐€๐’ and ๐€๐’

๐Ÿ

-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]

Abstractโ€”The purpose of this paper is to introduce the two new concepts of ๐‘จ๐‘บ-Paracompact spaces and ๐‘จ๐‘บ๐Ÿ-Paracompact spaces. Also we have proved that every ๐‘จ๐‘บ-Paracompactness and ๐‘จ๐‘บ๐Ÿ-Paracompactness has a topological property. Furthermore, we have introduced ๐€๐’-normal and its properties.

Keywordsโ€”Angelic spaces, seperable,๐‘บ-Paracompact,๐‘บ๐Ÿโ€“Paracompact, ๐‘บ-Normal, ๐‘จ๐‘บ- Paracompact, ๐‘จ๐‘บ๐Ÿโ€“Paracompact, ๐‘จ๐‘บ-normal.

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

I. INTRODUCTION

In this paper, we introduce two new properties in topological spaces which are ๐ด๐‘†- paracompactness and ๐ด๐‘†2-paracompactness. And also introduced the concept of ๐ด๐‘†-normal.

Diedonne.Jintroduced by [4] Paracompact space. Arhangelskii defined by๐ถ-paracompact and ๐ถ2-paracompact.๐ถ-paracompact and ๐ถ2-paracompact were studied in [15]. Arhangelskii introduced on 2012 a new weaker version of normality issaid to ๐ถ-normality [2]. In 2019, ๐‘†- paracompactness and ๐‘†2-paracompactness were studied in [13] and Lutfi Kalantan [8]

introduced ๐‘†-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 ๐‘†-paracompact, ๐‘†2-paracompact and ๐‘†-normal. Angelic paracompact space represented by ๐’œ๐’ซ๐’ฎ and angelic paracompact denoted by ๐’œ๐’ซ.

II. PRELIMINARIES Definition 2.1 :[3]

A ๐‘‡๐‘† is said to bean angelic, if for each relatively countably compact subset ๐พ of ๐น the following hold: (a) ๐พ is relatively compact (b) If ๐‘˜ โˆˆ ๐พ, then โˆƒa sequence in ๐พ that converges to ๐‘˜.

Definition 2.2:[12]

A ๐‘‡๐‘† is termed asseperable if it has a countable dense subset.

Definition 2.3:[13]

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Let ๐น be a TS and ๐พ be an angelic seperable subspace of ๐น. If there is a bijection mapping ๐‘Ÿ: ๐น โ†’ ๐บ, ๐บ is an paracompact(Hausdorff paracompact) space and the restriction ๐‘Ÿ ๐‘˜ : ๐พ โ†’ ๐‘Ÿ(๐พ) is a homeomorphism, then ๐น is said to be a๐‘บ-paracompact(๐‘บ๐Ÿ-paracompact) space.

Definition 2.4:[11]

ATS๐น is termed asa P-space if ๐น is ๐‘‡1 and each๐บ๐›ฟ-set is open.

Definition 2.5:[6]

A TS๐น is said to be locally seperable, if each element has a seperable open neighbourhood.

Definition 2.6:[14]

A space ๐น is of countable tightness if for every subset ๐พ of ๐น and every๐‘“ โˆˆ ๐น with ๐‘“ โˆˆ ๐พโˆƒ a countable subset ๐ฟ โŠ† ๐น โˆ‹ ๐‘™ โˆˆ ๐ฟ

III. ๐‘จ๐‘บ-PARACOMPACTNESS AND ๐‘จ๐‘บ๐Ÿ-PARACOMPACTNESS

Definition 3.1:

Let ๐‘‹ be an angelic space and ๐ด be an angelic seperable subspace of ๐‘‹. If โˆƒ a bijection mapping ๐‘“: ๐‘‹ โ†’ ๐‘Œ,๐‘Œ is an ๐’œ๐’ซ๐’ฎ and the restriction ๐‘” ๐’œ: ๐’œ โ†’ ๐‘”(๐’œ) is a homeomorphism, then ๐’ซ is said to be an ๐‘จ๐‘บ-paracompact space.

Definition 3.2:

Let ๐’ซ be an angelic space and ๐’œ be an angelic seperable subspace of ๐’ซ. If there is a bijection mapping ๐‘”: ๐’ซ โ†’ ๐’ฌ, ๐’ฌ is Hausdorff ๐’œ๐’ซ๐’ฎ and the restriction ๐‘” ๐’œ: ๐’œ โ†’ ๐‘”(๐’œ) is a homeomorphism, then ๐’ซ is said to be an ๐‘จ๐‘บ๐Ÿ-paracompact space.

Theorem 3.3:

Let๐น be an angelic separable but not ๐‘‡2, then ๐น cannot be an ๐ด๐‘†2-paracompact.

Proof.

Suppose๐น is any separable non-๐‘‡2 space. Let๐นbe an๐ด๐‘†2-paracompact. Then

โˆƒ a๐‘‡2๐’œ๐’ซ๐’ฎ๐บ and a bijective function ๐‘Ÿ: ๐น โ†’ ๐บ โˆ‹ ๐‘Ÿ ๐พ: ๐พ โ†’ ๐‘Ÿ(๐พ) is a homeomorphism for all angelic separable subspaces๐พ โŠ† ๐น. Since ๐น is separable, then ๐‘Ÿ: ๐น โ†’ ๐บ is a homeomorphism.

But ๐บ is Hausdorff, then ๐น is Hausdorff which is a contradiction. Hence ๐น cannot be ๐ด๐‘†2- paracompact.

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Example 3.4:

Let โ„, ๐’žโ„ฑ be the finite complement topology defined on the real numbers. Since (โ„, ๐’žโ„ฑ) is an ๐’œ๐’ซ being an angelic compact, then the identity function ๐‘–๐‘‘: โ„, ๐’žโ„ฑ โ†’ (โ„, ๐’žโ„ฑ) shows that it is an๐ด๐‘†-paracompact but not an๐ด๐‘†2-paracompact being angelic separable but not ๐‘‡2 space.

Example 3.5.

ฮฉ1 is an ๐ด๐‘†2-paracompact space which is not an angelic paracompact. Any angelic separable subspace of ฮฉ1 is countable. Suppose ๐พ โŠ‚ฮฉ1 is uncountable which implies that ๐พ is unbounded in ฮฉ1. If ๐‘ is any countable subset of ๐พ, then โˆƒ ๐›ผ < ฮฉ1 โˆ‹ ๐›ผ = ๐‘†๐‘ข๐‘ ๐‘. Thus,

โˆƒ ๐œ‚ โˆˆ ๐พ โˆ‹ ๐›ผ < ๐œ‚ . The set ( ๐›ผ, ๐œ‚ โˆฉ ๐พ) is a nonempty open subset of ๐พ with ๐›ผ, ๐œ‚ โˆฉ ๐พ โˆฉ ๐‘ = โˆ…. Thus, ๐พ cannot be an angelic separable implying that any angelic separable subspace of ฮฉ1 is countable. Since ฮฉ1 is ๐‘‡2 locally angelic compact, then โˆƒ a one to one continuous function, say ๐‘Ÿ, onto a Hausdorff compact space ๐บ. Let ๐พ โŠ‚ฮฉ1 be any angelic separable subspace. Since the closure of any countable set is an angelic compact in ฮฉ1, then we get that ๐‘Ÿ ๐พ: ๐พ โ†’ ๐‘Ÿ(๐พ) is a homeomorphism implying ๐‘Ÿ ๐พ: ๐พ โ†’ ๐‘Ÿ(๐พ) is a homeomorphism.

Theorem 3.6:

If ๐น is an angelic separable but not an angelic paracompact, then ๐น cannot be an ๐ด๐‘†- paracompact.

Proof:

Suppose๐น is any angelic separable non-๐’œ๐’ซ๐’ฎ. Let๐น be an๐ด๐‘†-paracompact. Then

โˆƒ an๐’œ๐’ซ๐’ฎ๐บ and a bijective function ๐‘Ÿ: ๐น โ†’ ๐บ โˆ‹ ๐‘Ÿ ๐พ: ๐พ โ†’ ๐‘Ÿ(๐พ) is a homeomorphism for everyangelic separable subspaces๐พ โŠ† ๐น. Since ๐น is an angelic separable, then ๐‘Ÿ: ๐น โ†’ ๐บ is a homeomorphism. But ๐บ is an ๐’œ๐’ซ, then ๐น is ๐’œ๐’ซ which is a contradiction. Hence ๐น cannot be an ๐ด๐‘† -paracompact.

Theorem 3.7:

Every ๐‘‡2 countably compact angelic separable ๐ด๐‘† -paracompact space isan angelic compact.

Proof:

Suppose๐นis any ๐‘‡2 countably angelic compact angelic separable ๐ด๐‘† -paracompact space. Then ๐น is an ๐’œ๐’ซ because the witness function of ๐ด๐‘† -paracompactness is a homeomorphism. Since any countably angelic compact ๐‘‡2๐’œ๐’ซ๐’ฎ is an angelic compact.

Hence๐น is an angelic compact.

Theorem 3.8:

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If ๐น is an ๐ด๐‘† -paracompact (๐ด ๐‘†2-paracompact) space and Frยดechet such that ๐‘Ÿ โˆถ ๐น โ†’ ๐บ is a witness of ๐ด๐‘† -paracompactness (๐ด๐‘† 2-paracompactness) of ๐น, then ๐‘Ÿ is continuous.

Proof:

Suppose๐น is an๐ด๐‘† -paracompact and Frechet. Let ๐‘Ÿ โˆถ ๐น โ†’ ๐บ witnesses ๐ด๐‘† - paracompactness of ๐น. Let ๐พ โŠ† ๐น and pick ๐‘” โˆˆ ๐‘Ÿ (๐พ ) implying that โˆƒ ๐‘Ž a unique ๐‘“ โˆˆ ๐น โˆ‹ ๐‘Ÿ (๐‘“ ) = ๐‘” and ๐‘“ โˆˆ ๐พ. Since๐น is Frechet, then โˆƒa sequence ๐‘Ž๐‘› โŠ† ๐พ โˆ‹ ๐‘Ž๐‘› โ†’๐‘“. The subspace ๐ฟ = {๐‘“ , ๐‘Ž๐‘› โˆถ ๐‘› โˆˆ ๐‘} of ๐น is an angelic separable by being countable, thus ๐‘Ÿ ๐ฟ: ๐ฟ โ†’ ๐‘Ÿ (๐ฟ ) is a homeomorphism. Now, let ๐‘ƒ โŠ† ๐บ be any open neighborhood of ๐‘”. Then ๐‘ƒ โˆฉ ๐‘Ÿ (๐ฟ ) is open in the subspace ๐‘Ÿ ๐ฟ โŠ† ๐‘”. Since ๐‘Ÿ ({๐‘Ž๐‘› โˆถ ๐‘› โˆˆ ๐‘}) โŠ† ๐‘Ÿ (๐ฟ ) โˆฉ ๐‘Ÿ (๐พ) and ๐‘ƒ โˆฉ ๐‘Ÿ ๐ฟ โ‰  โˆ… , ๐‘ƒ โˆฉ ๐‘Ÿ ๐ฟ โ‰  โˆ… . Hence ๐‘” โˆˆ ๐‘Ÿ (๐พ) , thus ๐‘Ÿ (๐พ ) โŠ† ๐‘Ÿ (๐ด ) . Therefore, ๐‘Ÿ is continuous.

Theorem 3.9:

๐ด๐‘† -paracompactness (๐ด ๐‘†2-paracompactness) is a topological property.

Proof:

Let ๐น be an ๐ด๐‘† -paracompact space and ๐ป be any TSโˆ‹๐น is homeomorphic to ๐ป. Let ๐‘Ÿ be the function witnessing ๐ด๐‘† -paracompactness of ๐น onto an ๐’œ๐’ซ๐’ฎ๐บ and ๐‘ก โˆถ ๐น โ†’ ๐บ be a homeomorphism. Then ๐‘Ÿ โˆ˜ ๐‘ก โˆ’1 โˆถ ๐ป โ†’ ๐บ will be the witness of ๐ด๐‘† -paracompactness of ๐ป. Hence ๐ด๐‘† -paracompactness (๐ด ๐‘†2-paracompactness) is a topological property.

Theorem 3.10:

The Cartesian product of two ๐ด ๐‘†2-paracompact spaces is ๐ด ๐‘†2-paracompact in case that at least one of them is countably compact and Frยดechet.

Proof:

Let ๐น and ๐ป be ๐ด๐‘† 2-paracompact โˆ‹ ๐น is countably compact and Frechet. Let๐บ and ๐‘Ÿ โˆถ ๐น โ†’ ๐บ be witnesses of ๐ด ๐‘†2-paracompactness of ๐น. Then ๐‘Ÿ is continuous by Theorem 3.8 implying that๐บ is countably angelic compact. Hence, ๐บ is compact. Let ๐บโ€ฒ and ๐‘Ÿ โ€ฒโˆถ ๐ป โ†’ ๐บโ€ฒ be witnesses of ๐ด ๐‘†2-paracompactness of ๐ป. Consider the function ๐‘ก โˆถ= ๐‘Ÿ ร— ๐‘Ÿ โ€ฒ: ๐น ร— ๐ป โ†’ ๐บ ร— ๐บ โ€ฒ . Observe that ๐บ ร— ๐บโ€ฒ is ๐‘‡2angelic paracompact. Now, let ๐‘ be any angelic separable subspace of ๐น ร— ๐ป. Therefore, ๐‘1(๐‘) โŠ† ๐น and ๐‘2(๐‘) โŠ† ๐ป are both separable subspaces of ๐น and๐ป respectively being continuous images of an angelic separable subspace ๐‘ โŠ† ๐น ร— ๐ป. Then countable product of an angelic separable spaces is angelic separable, ๐‘1(๐‘) ร— ๐‘2(๐‘) is an angelic separable in ๐น ร— ๐ป. Thus, as ๐‘ โŠ† ๐‘1(๐‘) ร— ๐‘2(๐‘), we get that ๐‘ก ๐ฟ: ๐ฟ โ†’ ๐‘ก (๐ฟ ) is a homeomorphism. Hence the Cartesian product of two ๐ด ๐‘†2-paracompact spaces is ๐ด ๐‘†2-paracompact in case that at least one of them is countably compact and Frยดechet.

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Theorem 3.11:

๐ด๐‘† -paracompactness (๐ด ๐‘†2-paracompactness) is an additive property.

Proof:

Suppose {๐น๐›ผ โˆถ ๐›ผ โˆˆ ๐›ฌ } is a family of ๐ด๐‘† 2 -paracompact spaces. Hence, for every๐›ผ โˆˆ ๐›ฌ โˆƒan๐’œ๐’ซ๐’ฎ ๐บ๐›ผ and a bijection ๐‘Ÿ๐›ผ โˆถ ๐น๐›ผ โ†’ ๐บ๐›ผ โˆ‹๐‘Ÿ๐›ผ ๐พ๐›ผ โˆถ ๐พ๐›ผ โ†’ ๐‘Ÿ๐›ผ(๐พ๐›ผ) is a homeomorphism for everyangelic separable subspace ๐พ๐›ผ of ๐น๐›ผ. Since angelic paracompactness is an additive property,โŠ•๐›ผโˆˆ๐›ฌ ๐บ๐›ผ is an angelic paracompact. Define ๐‘Ÿ โˆถ โŠ•๐›ผโˆˆ๐›ฌ ๐น โ†’ โŠ•๐›ผโˆˆ๐›ฌ ๐บ๐›ผ as follows: for each ๐‘“ โˆˆโŠ•๐›ผโˆˆ๐›ฌ ๐น๐›ผ, there exists a unique ๐›พ โˆˆ ๐›ฌ such that ๐‘“ โˆˆ ๐น๐›พ then ๐‘Ÿ (๐‘“ ) = ๐‘Ÿ๐›พ(๐‘“ ). Let ๐พ be any angelic separable subspace of

โŠ•๐›ผโˆˆ๐›ฌ ๐น๐›ผ. Write ๐พ = โˆช ๐›ผ โˆˆ ๐›ฌโˆ— (๐พ โˆฉ ๐น๐›ผ) where ๐›ฌโˆ— = {๐›ผ โˆˆ ๐›ฌ โˆถ ๐พ โˆฉ ๐น๐›ผ ,โˆ… }. Since ๐พ is an angelic separable, then ๐›ฌโˆ— is countable and ๐พ โˆฉ ๐น๐›ผis an angelic separable in ๐น๐›ผ for all ๐›ผ โˆˆ ๐›ฌโˆ—. Therefore, ๐‘Ÿ๐›ผ ๐พโˆฉ๐‘‹๐›ผ โˆถ ๐พ โˆฉ ๐น๐›ผ โ†’ ๐‘Ÿ๐›ผ(๐พ โˆฉ ๐น๐›ผ) is a homeomorphism for every๐›ผ โˆˆ ๐›ฌโˆ— implying that ๐‘Ÿ ๐พ: ๐พ โ†’ ๐‘Ÿ (๐พ) is a homeomorphism.

Theorem3.12:

Every second countable ๐ด ๐‘†2-paracompact space is metrizable.

Proof:

Suppose(๐น , ๐œ ) is an ๐ด ๐‘†2-paracompact second countable space which yields that ๐น is an angelic separable ๐ด๐‘† 2-paracompact. Then ๐น is ๐‘‡4 implying that ๐น is regular. Since any second countable ๐‘‡3 space is metrizable, we get that ๐น is metrizable.

Theorem 3.13:

Every P-space is ๐ด๐‘† 2-paracompact.

Proof:

Suppose(๐น , ๐œ ) is a P-space. If ๐น is countable, then it is discrete implying that ๐น is an๐ด๐‘† 2-paracompact. Assume ๐น is uncountable. Let ๐พ โŠ† ๐น be an arbitrary uncountable subset of ๐น and let ๐‘ โŠ‚ ๐พ be any countable subset of ๐พ. Then ๐‘ is closed set in ๐พ and ๐พ \๐‘ is a non-empty open set in ๐พ with (๐พ \๐‘) โˆฉ ๐‘ = โˆ…. Hence, ๐‘ cannot be dense in ๐พ implying that ๐พ cannot be separable. Thus, any angelic separable subspace of ๐น must be countable and hence any angelic separable subspace of ๐น is discrete. Take the identity map ๐‘–๐‘‘ โˆถ (๐น , ๐œ ) โ†’ (๐น , ๐‘). Then ๐‘–๐‘‘ ๐พ: ๐พ โ†’ ๐‘Ÿ (๐พ) is a homeomorphism for everyangelic separable subspaces ๐พ. Hence, (๐น , ๐œ ) is an๐ด๐‘† 2-paracompact.

Example 3.14:

Let(โ„,๐ถ๐ถ ), where ๐ถ๐ถ is the countable complement topology defined on โ„. Since (โ„,๐ถ๐ถ ) is P-space, then by Theorem 3.13, (โ„,๐ถ๐ถ ) is ๐ด๐‘† 2-paracompact. Note function witnessing the ๐ด๐‘† 2-paracompactness here is the identity taken from ๐ถ๐ถ to the discrete topology defined on โ„, write (โ„,๐ท). However, ๐‘–๐‘‘ โˆถ (โ„,๐ถ๐ถ ) โ†’ (โ„,๐ท) is not continuous.

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IV. ๐‘จ๐‘บ -NORMAL AND ITS PROPERTIES

Definition 4.1:

A ๐‘‡๐‘†๐น is termed as an๐ด๐‘† -normal if โˆƒ a normal space ๐บ and a bijective function ๐‘Ÿ โˆถ ๐น โ†’ ๐บ โˆ‹ the restriction ๐‘Ÿ ๐พ โˆถ ๐พ โ†’ ๐‘Ÿ (๐พ) is a homeomorphism for everyangelic separable subspace ๐พ โŠ† ๐น.

Example 4.2:

Consider the left ray topology defined on โ„, (โ„,๐ฟ ) such that ๐ฟ = โˆ… ,โ„ โˆช { โˆ’โˆž,๐‘Ž : ๐‘Ž โˆˆ โ„}. It is an example of ๐ด๐‘† -normal space by being a normal space which is not ๐ด๐‘† 2-paracompact space since it is separable and not ๐‘‡2 space. In fact, it is not even an ๐ด๐‘† -paracompact because it is separable not an๐’œ๐’ซ๐’ฎ .

Theorem 4.3:

๐ด๐‘† -normality has a topological property.

Proof:

Suppose๐น is an ๐ด๐‘† -normal space and ๐น โ‰… ๐ป and๐บ is a normal space and ๐‘Ÿ โˆถ ๐น โ†’ ๐บ be a bijective function โˆ‹๐‘Ÿ ๐‘€ โˆถ ๐‘€ โ†’ ๐‘Ÿ (๐‘€) is a homeomorphism for everyangelic separable subspace ๐‘€ of ๐น. Let ๐‘ก โˆถ ๐ป โ†’ ๐น be a homeomorphism. Hence ๐บ and๐‘Ÿ โˆ˜ ๐‘ก โˆถ ๐ป โ†’ ๐บhas the topological property.

Theorem 4.4:

๐ด๐‘† -normality is an additive property.

Proof:

Suppose๐น๐›ผ be an ๐ด๐‘† -normal space for every๐›ผ โˆˆ ๐›ฌ. Then their sum โŠ•๐›ผโˆˆ๐›ฌ ๐น๐›ผis ๐ด๐‘† -normal. For every๐›ผ โˆˆ ๐›ฌ, pick a normal space ๐บ๐›ผ and a bijective function ๐‘Ÿ๐›ผ โˆถ ๐น๐›ผ โ†’ ๐บ๐›ผ โˆ‹๐‘Ÿ๐›ผ

๐‘€๐›ผ โˆถ ๐‘€๐›ผ โ†’ ๐‘Ÿ๐›ผ(๐‘€๐›ผ) is a homeomorphism for each angelic separable subspace ๐‘€๐›ผ of ๐น๐›ผ. Since ๐บ๐›ผ is normal for every๐›ผ โˆˆ ๐›ฌ, the sum โŠ•๐›ผโˆˆ๐›ฌ ๐บ๐›ผ is normal. Consider the function sum, โŠ•๐›ผโˆˆ๐›ฌ ๐‘Ÿ๐›ผ:โŠ•๐›ผโˆˆ๐›ฌ ๐น๐›ผ โ†’โŠ•๐›ผโˆˆ๐›ฌ ๐บ๐›ผ defined by โŠ•๐›ผโˆˆ๐›ฌ ๐‘Ÿ๐›ผ(๐‘“ ) = ๐‘Ÿ ๐›ฝ (๐‘“ ) if ๐‘“ โˆˆ ๐น๐›ฝ, ๐›ฝ โˆˆ ๐›ฌ. Now, a subspace ๐‘€ โŠ†โŠ•๐›ผโˆˆ๐›ฌ ๐น๐›ผis an angelic separable if and only if the set ๐›ฌ0 = {๐›ผ โˆˆ ๐›ฌ โˆถ ๐‘€ โˆฉ ๐น๐›ผ โ‰  โˆ… } is countable and ๐‘€ โˆฉ ๐น๐›ผ is angelic separable in ๐น๐›ผ for each ๐›ผ โˆˆ ๐›ฌ0 . If ๐‘€ โŠ†โŠ•๐›ผโˆˆ๐›ฌ ๐น๐›ผ is an angelic separable, then โŠ•๐›ผโˆˆ๐›ฌ ๐‘Ÿ๐›ผ ๐‘€ is a homeomorphism because ๐‘Ÿ๐›ผ ๐‘€ โˆฉ ๐น๐›ผ is a homeomorphism for every๐›ผ โˆˆ ๐›ฌ0.

Theorem 4.5:

If ๐น is an๐ด๐‘† -normal and of countable tightness and ๐‘Ÿ โˆถ ๐น โ†’ ๐บ witnesses the ๐ด๐‘† - normality of ๐น. Then ๐‘Ÿ is continuous.

Proof:

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Suppose that ๐น is an ๐ด๐‘† -normal and of countable tightness. Let ๐‘Ÿ โˆถ ๐น โ†’ ๐บ be any witness of the ๐ด๐‘† -normality of ๐น . Let ๐พ โŠ† ๐น be arbitrary. We have ๐‘Ÿ (๐พ ) = ๐‘Ÿ ( ๐ฟโˆˆ ๐พ โ‰คฮฉ0๐ฟ ) = ๐ฟโˆˆ ๐พ โ‰คฮฉ0๐‘Ÿ ๐ฟ โŠ† ๐ฟโˆˆ ๐พ โ‰คฮฉ ๐‘Ÿ (๐ฟ )

0 โŠ† ๐‘Ÿ (๐พ) . Therefore, ๐‘Ÿ is continuous.

Remark:

Any first countable space is Fr๐‘’ chet, any Fr๐‘’ chet space is sequential and any sequential space is of countable tightness in [6] we conclude the following corollary.

Corollary 4.6:

(1) If ๐น is ๐ด๐‘† -normal first countable and ๐‘Ÿ โˆถ ๐น โ†’ ๐บ witnesses the ๐ด๐‘† -normality of ๐น. Then ๐‘Ÿ is continuous.

(2) If ๐น is ๐ด๐‘† -normal Fr๐‘’ chet and ๐‘Ÿ โˆถ ๐น โ†’ ๐บ witnesses the ๐ด๐‘† -normality of ๐น. Thenr๐‘“ is continuous.

(3) If ๐น is ๐ด๐‘† -normal sequential and ๐‘Ÿ โˆถ ๐น โ†’ ๐บ witnesses the ๐ด๐‘† -normality of ๐น. Then ๐‘Ÿ is continuous.

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