scholarly journals Characterizations of Strongly Paracompact Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Xin Zhang
Keyword(s):  
Open Cover ◽  
Normal Space ◽  
Compact Spaces ◽  
Countably Paracompact ◽  
Star Countable ◽  
Perfectly Normal Space ◽  
Paracompact Spaces ◽  
Perfectly Normal

Characterizations of strongly compact spaces are given based on the existence of a star-countable open refinement for every increasing open cover. It is proved that a countably paracompact normal space (a perfectly normal space or a monotonically normal space) is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement. Moreover, it is shown that a space is linearlyDprovided that every increasing open cover of the space has a point-countable open refinement.

Countably Compact Spaces and Martin's Axiom

1978 ◽  
Vol 30 (02) ◽  
pp. 243-249 ◽  
Author(s):  
William Weiss
Keyword(s):  
Set Theory ◽  
Normal Space ◽  
Topological Spaces ◽  
Compact Spaces ◽  
Countably Compact Space ◽  
Countably Compact ◽  
Countably Compact Spaces ◽  
Perfectly Normal Space ◽  
The Relationship ◽  
Perfectly Normal

The relationship between compact and countably compact topological spaces has been studied by many topologists. In particular an important question is: “What conditions will make a countably compact space compact?” Conditions which are “covering axioms” have been extensively studied. The best results of this type appear in [19]. We wish to examine countably compact spaces which are separable or perfectly normal. Recall that a space is perfect if and only if every closed subset is a Gδ, and that a space is perfectly normal if and only if it is both perfect and normal. We show that the following statement follows from MA +┐ CH and thus is consistent with the usual axioms of set theory: Every countably compact perfectly normal space is compact. This result is Theorem 3 and can be understood without reading much of what goes before.

Sum Theorems for Countably Paracompact Spaces

1973 ◽  
Vol 25 (4) ◽  
pp. 706-711
Author(s):  
Henry Potoczny
Keyword(s):  
Open Cover ◽  
Equivalent Formulation ◽  
Countably Paracompact ◽  
Definition Of ◽  
Paracompact Spaces ◽  
Countable Paracompactness

In this paper, we extend the class of spaces to which the Σ and β theorems of Hodel apply, as well as the sum and subset theorems of [2]. Instead of the open cover definition of countable paracompactness, we utilize an equivalent formulation of countable paracompactness, due to Ishikawa [3].

scholarly journals PAIRS OF HAHN AND SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 210-229
Author(s):  
O. Maslyuchenko ◽  
A. Kushnir
Keyword(s):  
Continuous Function ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Normal Space ◽  
Semicontinuous Function ◽  
Upper Semicontinuous ◽  
Pseudocompact Space ◽  
Upper Semicontinuous Function ◽  
Perfectly Normal Space ◽  
Perfectly Normal

In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the infimum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactification any pair of Hahn on X is generated by a separately continuous function on X x Y .

scholarly journals A characterization of the uniform convergence points set of some convergent sequence of functions

2021 ◽  
Vol 71 (2) ◽  
pp. 423-428
Author(s):  
Olena Karlova
Keyword(s):  
Uniform Convergence ◽  
Convergent Sequence ◽  
Normal Space ◽  
Disjoint Sequence ◽  
Isolated Points ◽  
Perfectly Normal Space ◽  
Set Of Points ◽  
Sequence Of Functions ◽  
Perfectly Normal

Abstract We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A ⊆ X, then A is the set of points of the uniform convergence for some convergent sequence (fn ) n∈ω of functions fn : X → ℝ if and only if A is Gδ -set which contains all isolated points of X. This result generalizes a theorem of Ján Borsík published in 2019.

An example of a hereditarily normal topologically finite space, which is topologically infinite relative to the class of all its proper F σ-subspaces

2019 ◽  
Vol 26 (2) ◽  
pp. 315-319
Author(s):  
Ivane Tsereteli
Keyword(s):  
Normal Space ◽  
Finite Space ◽  
Perfectly Normal Space ◽  
Open Subspace ◽  
Perfectly Normal

Abstract A (Hausdorf) hereditarily normal (not perfectly normal) space X is constructed, which has the following properties: (a) there exists a proper open subspace of X which is homeomorphic to the whole X (i.e., the space X is topologically infinite); (b) the space is homeomorphic to none of its proper {F_{\sigma}} -subspaces (i.e., the space X is topologically finite relative to the class of all its proper {F_{\sigma}} -subspaces).

On star Lindelöf spaces

2020 ◽  
Vol 57 (2) ◽  
pp. 139-146
Author(s):  
Wei-Feng Xuan ◽  
Yan-Kui Song
Keyword(s):  
Hausdorff Space ◽  
Normal Space ◽  
Star Countable

AbstractIn this paper, we prove that if X is a space with a regular Gδ-diagonal and X2 is star Lindelöf then the cardinality of X is at most 2c. We also prove that if X is a star Lindelöf space with a symmetric g-function such that {g2(n, x): n ∈ ω} = {x} for each x ∈ X then the cardinality of X is at most 2c. Moreover, we prove that if X is a star Lindelöf Hausdorff space satisfying Hψ(X) = κ then e(X) 22κ; and if X is Hausdorff and we(X) = Hψ(X) = κsubset of a space then e(X) 2κ. Finally, we prove that under V = L if X is a first countable DCCC normal space then X has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in Spaces with property (DC(ω1)), Comment. Math. Univ. Carolin., 58(1) (2017), 131-135.

scholarly journals Paracompactness with respect to an ideal

1997 ◽  
Vol 20 (3) ◽  
pp. 433-442 ◽  
Author(s):  
T. R. Hamlett ◽  
David Rose ◽  
Dragan Janković
Keyword(s):  
Topological Space ◽  
Finite Union ◽  
Open Cover ◽  
Locally Finite ◽  
Open Set ◽  
Special Cases ◽  
Paracompact Spaces

An ideal on a setXis a nonempty collection of subsets ofXclosed under the operations of subset and finite union. Given a topological spaceXand an idealℐof subsets ofX,Xis defined to beℐ-paracompact if every open cover of the space admits a locally finite open refinement which is a cover for all ofXexcept for a set inℐ. Basic results are investigated, particularly with regard to theℐ- paracompactness of two associated topologies generated by sets of the formU−IwhereUis open andI∈ℐand⋃{U|Uis open andU−A∈ℐ, for some open setA}. Preservation ofℐ-paracompactness by functions, subsets, and products is investigated. Important special cases ofℐ-paracompact spaces are the usual paracompact spaces and the almost paracompact spaces of Singal and Arya [“On m-paracompact spaces”, Math. Ann., 181 (1969), 119-133].

On Countably Paracompact Normal Spaces

1957 ◽  
Vol 9 ◽  
pp. 443-449 ◽  
Author(s):  
M. J. Mansfield
Keyword(s):  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normal Space ◽  
Countably Paracompact ◽  
Necessary And Sufficient

A. H. Stone (9), E. Michael (3, 4), J. L. Kelley and J. S. Griff en (2) have established many necessary and sufficient conditions that a regular Hausdorff space be paracompact. It is the purpose of this note to show that if the word “countable” is inserted in the appropriate places in the above-mentioned conditions they become, in general, necessary and sufficient conditions that a normal space be countably paracompact.

scholarly journals On normal covers of locally compact spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 343-347
Author(s):  
Yukinobu Yajima
Keyword(s):  
Open Cover ◽  
Locally Compact ◽  
Compact Closure ◽  
Compact Spaces ◽  
Locally Compact Spaces

In this paper, we deal with the following question: What kind of open covers are normal if they have cushioned open refinements? For this, we prove that an open cover consisting of members with compact closure is a desired one.

scholarly journals PRODUCTS OF BASE-κ-PARACOMPACT SPACES AND COMPACT SPACES

2011 ◽  
Vol 84 (3) ◽  
pp. 387-392
Author(s):  
LEI MOU
Keyword(s):  
Compact Spaces ◽  
Paracompact Spaces ◽  
Regular Ordinal

AbstractLet λ be a regular ordinal with λ≥ω1. Then we prove that (λ+1)×λ is not base-countably metacompact. This implies that base-κ-paracompactness is not an inverse invariant of perfect mappings, which answers a question asked by Yamazaki.

Sign in / Sign up

Export Citation Format

Share Document