Locally Compact Spaces

1950 ◽  
pp. 216-249
Author(s):  
Paul R. Halmos
Keyword(s):  
Locally Compact ◽  
Compact Spaces ◽  
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.

About Weakly Locally Compact Spaces

2004 ◽  
pp. 638-644
Author(s):  
Tomas K. Breuckmann ◽  
Soraya R. T. Kudri ◽  
Halis Aygün
Keyword(s):  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Spaces

A Best Possible Tauberian Theorem for the Collective Continuous Hausdorff Summability Method

1971 ◽  
Vol 23 (3) ◽  
pp. 544-549
Author(s):  
G. E. Peterson
Keyword(s):  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Bounded Complex ◽  
Hausdorff Method ◽  
Borel Functions ◽  
Bounded Functions ◽  
Abelian Theorem ◽  
Complex Valued ◽  
Locally Compact Spaces

The purpose of this paper is to prove that o(l/x) is the best possible Tauberian condition for the collective continuous Hausdorff method of summation. The analogue of this result for the collective (discrete) Hausdorff method is known [1, pp. 229, ff.; 7, p. 318; 8, p. 254]. Our method involves generalizing a well-known Abelian theorem of Agnew [2] to locally compact spaces and then applying the analogue for integrals of a result Lorentz obtained for series [6, Theorem 1].Let T and X denote locally compact, non compact, σ-compact Hausdorff spaces. Let T′ = T ∪ (∞) and X′ = X ∪ (∞) denote the onepoint compactifications of T and X, respectively. Let B(T) denote the set of locally bounded, complex valued Borel functions on T and let B∞(T) denote the bounded functions in B(T).

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