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On ω -resolvable and almost- ω -resolvable spaces

J. Angoa, M. Ibarra, Angel Tamariz-Mascarúa (2008)

Commentationes Mathematicae Universitatis Carolinae

We continue the study of almost- ω -resolvable spaces beginning in A. Tamariz-Mascar’ua, H. Villegas-Rodr’ıguez, Spaces of continuous functions, box products and almost- ω -resoluble spaces, Comment. Math. Univ. Carolin. 43 (2002), no. 4, 687–705. We prove in ZFC: (1) every crowded T 0 space with countable tightness and every T 1 space with π -weight ≤ ℵ 1 is hereditarily almost- ω -resolvable, (2) every crowded paracompact T 2 space which is the closed preimage of a crowded Fréchet T 2 space in such a way that the...

P λ -sets and skeletal mappings

Aleksander Błaszczyk, Anna Brzeska (2013)

Colloquium Mathematicae

We prove that if the topology on the set Seq of all finite sequences of natural numbers is determined by P λ -filters and λ ≤ , then Seq is a P λ -set in its Čech-Stone compactification. This improves some results of Simon and of Juhász and Szymański. As a corollary we obtain a generalization of a result of Burke concerning skeletal maps and we partially answer a question of his.

Partial dcpo’s and some applications

Zhao Dongsheng (2012)

Archivum Mathematicum

We introduce partial dcpo’s and show their some applications. A partial dcpo is a poset associated with a designated collection of directed subsets. We prove that (i) the dcpo-completion of every partial dcpo exists; (ii) for certain spaces X , the corresponding partial dcpo’s of continuous real valued functions on X are continuous partial dcpos; (iii) if a space X is Hausdorff compact, the lattice of all S-lower semicontinuous functions on X is the dcpo-completion of that of continuous real valued...

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

V. V. Tkachuk (2005)

Fundamenta Mathematicae

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 it possible...

Preimages of Baire spaces

Jozef Doboš, Zbigniew Piotrowski, Ivan L. Reilly (1994)

Mathematica Bohemica

A simple machinery is developed for the preservation of Baire spaces under preimages. Subsequently, some properties of maps which preserve nowhere dense sets are given.

Properties of Λ , δ -closed sets in topological spaces

D. N. Georgiou, S. Jafari, T. Noiri (2004)

Bollettino dell'Unione Matematica Italiana

In questo articolo vengono presentate e studiate le nozioni di insieme Λ δ e di insieme Λ , δ -chiuso. Inoltre, vengono introdotte le nozioni di Λ , δ -continuità, Λ , δ -compatezza e Λ , δ -connessione e vengono fornite alcune caratterizzazioni degli spazi δ - T 0 e δ - T 1 . Infine, viene mostrato che gli spazi Λ , δ -connessi e Λ , δ -compatti vengono preservati mediante suriezioni δ -continue.

Reflecting topological properties in continuous images

Vladimir Tkachuk (2012)

Open Mathematics

Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin...

Reflexive families of closed sets

Zhongqiang Yang, Dongsheng Zhao (2006)

Fundamenta Mathematicae

Let S(X) denote the set of all closed subsets of a topological space X, and C(X) the set of all continuous mappings f:X → X. A family 𝓐 ⊆ S(X) is called reflexive if there exists ℱ ⊆ C(X) such that 𝓐 = {A ∈ S(X): f(A) ⊆ A for every f ∈ ℱ}. We investigate conditions ensuring that a family of closed subsets is reflexive.

Relations approximated by continuous functions in the Vietoris topology

L'. Holá, R. A. McCoy (2007)

Fundamenta Mathematicae

Let X be a Tikhonov space, C(X) be the space of all continuous real-valued functions defined on X, and CL(X×ℝ) be the hyperspace of all nonempty closed subsets of X×ℝ. We prove the following result: Let X be a locally connected locally compact paracompact space, and let F ∈ CL(X×ℝ). Then F is in the closure of C(X) in CL(X×ℝ) with the Vietoris topology if and only if: (1) for every x ∈ X, F(x) is nonempty; (2) for every x ∈ X, F(x) is connected; (3) for every isolated x ∈ X, F(x) is a singleton...

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