Closed set countability axioms

Aull, C. E.

doi:10.1016/s1385-7258(66)50035-x

Cited by 14 publications

(7 citation statements)

References 5 publications

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“…Further, the identity map is closed and continuous, X itself is second countable, hence metrizable. Now Corollary 12, of [1], shows that the D x property characterizes the metric spaces which are the union of a compact set and isolated points. Since X has at most finitely many isolated points, X is compact metric, hence by Theorem 13, of [1], must be a D 2 space.…”

Section: Theorem 7 Let X Be a T 2 Space With At Most Finitely Many Imentioning

confidence: 99%

“…Aull introduced the Z>, and D 2 spaces in [1]. In this paper, we give a separation theorem for D { spaces, and provide a characterization of D, and D 2 spaces in terms of maps.…”

Section: Introductionmentioning

confidence: 99%

“…In [1], Aull presents a number of results about £>,, T y spaces. These can be strengthened slightly by replacing T 3 with T 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Thus, any closed continuous image of X is countable and first countable, hence second countable. But X is not D 2 , for as Aull shows in Theorem 13, of [1], D 2 metric space are compact.…”

Section: Introductionmentioning

confidence: 99%

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Maps on D₁ and D₂ spaces

Potoczny

1984

J Aust Math Soc A

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A space X is said to be Z), provided each closed set has a countable basis for the open sets containing it. It is said to be D 2 provided there is a countable base {(/"} such that each closed set has a countable base for the open sets containing it, which is a subfamily of {£/"}. In this paper, we give a separation theorem for D x spaces, and provide a characterization of D, and D 2 spaces in terms of maps.

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Section: Theorem 7 Let X Be a T 2 Space With At Most Finitely Many Imentioning

confidence: 99%

“…Aull introduced the Z>, and D 2 spaces in [1]. In this paper, we give a separation theorem for D { spaces, and provide a characterization of D, and D 2 spaces in terms of maps.…”

Section: Introductionmentioning

confidence: 99%

“…In [1], Aull presents a number of results about £>,, T y spaces. These can be strengthened slightly by replacing T 3 with T 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Thus, any closed continuous image of X is countable and first countable, hence second countable. But X is not D 2 , for as Aull shows in Theorem 13, of [1], D 2 metric space are compact.…”

Section: Introductionmentioning

confidence: 99%

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Maps on D₁ and D₂ spaces

Potoczny

1984

J Aust Math Soc A

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“…Let X be a space which is not almost compact. Then X can be embedded in a space Y such that: (1) X is not C*-embedded in F, (2) for x e X, ^(x, X) = \l,(x, F) and (3) for y Ç Y\X, t(y, F) = co.…”

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confidence: 99%

Absolute C-Embedding of Spaces with Countable Character and Pseudocharacter Conditions

Dow

1980

Can. j. math.

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Absolute C-embeddings have been studied extensively by C. E. Aull. We will use his notation P = C[Q] to mean that a space satisfying property Q is C-embedded in every space having property Q that it is embedded in if (and only if) it has property P. The first result of this type is due to Hewitt [5] where he proves that if Q is “Tychonoff” then P is almost compactness. Aull [2] proves that if Q is “T4 and countable pseudocharacter” or “T4 and first countable” then P is “countably compact”. In this paper we show that P is almost compactness if Q is “Tychonoff” and any of countable pseudocharacter, perfect, or first countability. Unfortunately for the last case we require the assumption that . Finally we show that P is countable compactness if Q is Tychonoff and “closed sets have a countable neighborhood base”. In each of the above results C-embedding may be replaced by C*-embeddings and the results hold if restricted to closed embeddings.

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On $$\mathfrak{M}$$ -paracompact spaces

Singal

Arya

1969

Math. Ann.

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The implications between different classes of topological spaces considered in this paper are summarized in the following diagram: cC Ace ~ AcPC cPCHere P = Para, C = Compact, c = Countably, A = Almost. The class of almost-gR-compact spaces has been introduced and studied by Singal and Singal [24].The authors are grateful to the referee for his valuable suggestions for the improvement of the paper.

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Closed set countability axioms

Cited by 14 publications

References 5 publications

Maps on D₁ and D₂ spaces

Maps on D₁ and D₂ spaces

Absolute C-Embedding of Spaces with Countable Character and Pseudocharacter Conditions

On $$\mathfrak{M}$$ -paracompact spaces

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Closed set countability axioms

Cited by 14 publications

References 5 publications

Maps on D1 and D2 spaces

Maps on D1 and D2 spaces

Absolute C-Embedding of Spaces with Countable Character and Pseudocharacter Conditions

On $$\mathfrak{M}$$ -paracompact spaces

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Maps on D₁ and D₂ spaces

Maps on D₁ and D₂ spaces