Displaying similar documents to “On k -spaces and k R -spaces”

A note on transitively D -spaces

Liang-Xue Peng (2011)

Czechoslovak Mathematical Journal

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In this note, we show that if for any transitive neighborhood assignment φ for X there is a point-countable refinement ℱ such that for any non-closed subset A of X there is some V ∈ ℱ such that | V ∩ A | ≥ ω , then X is transitively D . As a corollary, if X is a sequential space and has a point-countable w c s * -network then X is transitively D , and hence if X is a Hausdorff k -space and has a point-countable k -network, then X is transitively D . We prove that if X is a countably compact sequential space and...

k -systems, k -networks and k -covers

Jinjin Li, Shou Lin (2006)

Czechoslovak Mathematical Journal

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The concepts of k -systems, k -networks and k -covers were defined by A. Arhangel’skiǐ in 1964, P. O’Meara in 1971 and R. McCoy, I. Ntantu in 1985, respectively. In this paper the relationships among k -systems, k -networks and k -covers are further discussed and are established by m k -systems. As applications, some new characterizations of quotients or closed images of locally compact metric spaces are given by means of m k -systems.

In search for Lindelöf C p ’s

Raushan Z. Buzyakova (2004)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that if X is a first-countable countably compact subspace of ordinals then C p ( X ) is Lindelöf. This result is used to construct an example of a countably compact space X such that the extent of C p ( X ) is less than the Lindelöf number of C p ( X ) . This example answers negatively Reznichenko’s question whether Baturov’s theorem holds for countably compact spaces.

Point-countable π-bases in first countable and similar spaces

V. V. Tkachuk (2005)

Fundamenta Mathematicae

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It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable π-base. We look at general spaces with point-countable π-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable π-base. We also analyze when the function space C p ( X ) has a point-countable π -base, giving a criterion for this in terms of the topology of X when l*(X) = ω. Dealing with point-countable π-bases makes...

When is 𝐍 Lindelöf?

Horst Herrlich, George E. Strecker (1997)

Commentationes Mathematicae Universitatis Carolinae

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Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) ℕ is a Lindelöf space, (2) ℚ is a Lindelöf space, (3) ℝ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of ℝ is separable, (6) in ℝ , a point x is in the closure of a set A iff there exists a sequence in A that converges to x , (7) a function f : ℝ → ℝ is continuous at a point x iff f is sequentially continuous...