scholarly journals Locally compact perfectly normal spaces may all be paracompact

2010 ◽  
Vol 210 (3) ◽  
pp. 285-300 ◽  
Author(s):  
Paul B. Larson ◽  
Franklin D. Tall
Keyword(s):  
Locally Compact ◽  
Perfectly Normal

Locally Compact Normal Spaces in the Constructible Universe

1982 ◽  
Vol 34 (5) ◽  
pp. 1091-1096 ◽  
Author(s):  
W. Stephen Watson
Keyword(s):  
Set Theory ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Spaces ◽  
Perfectly Normal

Arhangel'skiĭ proved around 1959 [1] that, for the class of perfectly normal locally compact spaces, metacompactness and paracompactness are equivalent. It is shown to be consistent that this equivalence holds for the (larger) class of normal locally compact spaces (answering a question of Tall [8], [9]).The consistency of the existence of locally compact normal noncollectionwise Hausdorff spaces has been known since 1967. It is shown that the existence of such spaces is independent of the axioms of set theory, thus establishing that Bing's example G cannot be modified under ZFC to be locally compact.All topological spaces are assumed to be Hausdorff.First, a definition and three standard lemmata are needed.

Normal, Locally Compact, Boundedly Metacompact Spaces are Paracompact: an Application of Pixley-Roy Spaces

1983 ◽  
Vol 35 (5) ◽  
pp. 807-823 ◽  
Author(s):  
Peg Daniels
Keyword(s):  
Positive Integer ◽  
Partial Solution ◽  
Affirmative Answer ◽  
Open Cover ◽  
Locally Compact ◽  
Locally Finite ◽  
Finite Topology ◽  
Perfectly Normal

Let PR(X) denote the Pixley-Roy topology on the collection of all nonempty, finite subsets of a space X. For each cardinal κ, let κ* be the cardinal κ with the co-finite topology. We use PR(κ*) to obtain a partial solution in ZFC to F. Tall's question whether every normal, locally compact, metacompact space is paracompact [6]. W.S. Watson has answered this question affirmatively assuming V = L[7]. The question also has an affirmative answer if we assume either that the space is perfectly normal [1] or that it is locally connected [4].A space X is said to be boundedly metacompact (boundedly paracompact) provided that for each open cover of X there is a positive integer n such that has a point finite (locally finite) open refinement of order n. As the main result of this paper, we show every normal, locally compact, boundedly metacompact space is paracompact.

scholarly journals Countable sum theorem for locally closed sets

1984 ◽  
Vol 29 (1) ◽  
pp. 47-55
Author(s):  
Satya Deo ◽  
Dalip Singh Jamwal
Keyword(s):  
Cohomological Dimension ◽  
Locally Compact ◽  
Compact Spaces ◽  
Closed Sets ◽  
Sum Theorem ◽  
Locally Closed Sets ◽  
Locally Compact Spaces ◽  
Perfectly Normal

In this paper we show that the countable sum theorem for locally-closed sets, for sheaf theoretic cohomological dimension over a given ring L, holds in all perfectly normal spaces as well as in all locally compact spaces.

On 𝒞ℛ-Epic Embeddings and Absolute 𝒞ℛ-Epic Spaces

2005 ◽  
Vol 57 (6) ◽  
pp. 1121-1138 ◽  
Author(s):  
Michael Barr ◽  
R. Raphael ◽  
R. G. Woods
Keyword(s):  
Locally Compact ◽  
Compact Spaces ◽  
Open Questions ◽  
Tychonoff Spaces ◽  
Locally Compact Spaces

AbstractWe study Tychonoff spaces X with the property that, for all topological embeddings X → Y, the induced map C(Y ) → C(X) is an epimorphism of rings. Such spaces are called absolute 𝒞ℛ-epic. The simplest examples of absolute 𝒞ℛ-epic spaces are σ-compact locally compact spaces and Lindelöf P-spaces. We show that absolute CR-epic first countable spaces must be locally compact.However, a “bad” class of absolute CR-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute CR-epic abound, and some are presented.

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