A characterization of ℵ0-spaces

Guthrie, Joe A.

doi:10.1016/0016-660x(71)90116-4

Cited by 68 publications

(36 citation statements)

References 2 publications

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“…Consequently, every cosmic space (hence ℵ 0 -space as well) is Lindelöf. Another characterization of ℵ 0 -spaces is given by Guthrie in [24]. It is known [33] that an ℵ 0 -space (even an ℵ-space) which is either first countable or locally compact is metrizable.…”

Section: Introductionmentioning

confidence: 99%

On topological properties of Fréchet locally convex spaces with the weak topology

Gabriyelyan

Ka̧kol

Kubzdela

et al. 2015

Topology and its Applications

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Elsevier Gabriyelyan, S.; Kakol, JM.; Kubzdela, A.; López Pellicer, M. (2015). On topological properties of Fréchet locally convex spaces. Topology and its Applications. 192 AbstractWe describe the topology of any cosmic space and any ℵ 0 -space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ(E, E ) are ℵ 0 -spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) 1 and satisfying the Heinrich density condition we prove that (E, σ(E, E )) is an ℵ 0 -space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain 1 , then (E, σ(E, E )) is an ℵ 0 -space if and only if E is separable. The last part of the paper studies the question: Which spaces (E, σ(E, E )) are ℵ 0 -spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a (DF )-space whose strong dual E is separable, then (E, σ(E, E )) is an ℵ 0 -space. Supplementing an old result of Corson we show that, for ǎ Cech-complete Lindelöf space X the following are equivalent: (a) X is Polish, (b) C c (X) is cosmic in the weak topology, (c) the weak * -dual of C c (X) is an ℵ 0 -space.

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Section: Introductionmentioning

confidence: 99%

On topological properties of Fréchet locally convex spaces with the weak topology

Gabriyelyan

Ka̧kol

Kubzdela

et al. 2015

Topology and its Applications

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“…One can show that a space X is fc-semistratifiable if and only if X has a a-cushioned pair-cs-network (cf. [2,6,8]). Hence any space with a a-locally finite fc-network is fc-semistratifiable.…”

Section: (A) If U E T Then U = U"en5(« U) (B) If U E ^And N E N Tmentioning

confidence: 99%

Sequential Coreflections of Stratifiable Spaces

Foged¹

1984

Proceedings of the American Mathematical Society

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Abstract. We prove that the sequential coreflection of a stratifiable space which is the union of countably many closed fc-spaces is stratifiable.Several years ago E. Michael [10] showed that among spaces which are the union of countably many closed fc-spaces, the presence of a a-locally finite fc-network allows the preservation of certain topological properties (normality, paracompactness, stratifiability) when forming k(X). In this brief note we show that with use of the Lemma below essentially the same proof will work in fc-semistratifiable spaces.Definitions. Let (X, &~) be a Hausdorff topological space. A function 5: N X3T-+ {A g &>(x): A is closed in X) is a semistratification [3] of X if the following hold.

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“…The convergence of {>>": n < ω} contradicts that W is a sequential neighborhood of x. [3], his proof of the countable case in [2] can be adapted to show (c) implies (d), and the same is true for (a) implies (b). It therefore suffices to show (d) implies (a).…”

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confidence: 90%

Characterizations of ℵ-spaces

Foged¹

1984

Pacific J. Math.

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A characterization of ℵ0-spaces

Cited by 68 publications

References 2 publications

On topological properties of Fréchet locally convex spaces with the weak topology

On topological properties of Fréchet locally convex spaces with the weak topology

Sequential Coreflections of Stratifiable Spaces

Characterizations of ℵ-spaces

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