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).

Essentials of General Topology

2021 ◽  
pp. 191-244
Author(s):  
Adel N. Boules
Keyword(s):  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Section 8 ◽  
Finite Products ◽  
The One ◽  
One Point Compactification ◽  
Locally Compact Spaces ◽  
Metrization Theorem

The first eight sections of this chapter constitute its core and are generally parallel to the leading sections of chapter 4. Most of the sections are brief and emphasize the nonmetric aspects of topology. Among the topics treated are normality, regularity, and second countability. The proof of Tychonoff’s theorem for finite products appears in section 8. The section on locally compact spaces is the transition between the core of the chapter and the more advanced sections on metrization, compactification, and the product of infinitely many spaces. The highlights include the one-point compactification, the Urysohn metrization theorem, and Tychonoff’s theorem. Little subsequent material is based on the last three sections. At various points in the book, it is explained how results stated for the metric case can be extended to topological spaces, especially locally compact Hausdorff spaces. Some such results are developed in the exercises.

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.

On Normed Algebras which Satisfy a Reality Condition

1961 ◽  
Vol 13 ◽  
pp. 675-682
Author(s):  
Silvio Aurora
Keyword(s):  
Banach Algebra ◽  
Compact Space ◽  
Banach Algebras ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Reality Condition ◽  
Compact Spaces ◽  
Locally Compact Space ◽  
Locally Compact Spaces

Various results exist which permit real Banach algebras satisfying some sort of “reality condition” to be identified with the algebra of all continuous real-valued functions on a suitable compact space (or with the algebra of all continuous real-valued functions that * Vanish at infinity“ on a suitable non-compact, locally compact space in case the algebra has no unit). (Terminology follows (4), so that compact and locally compact spaces must be Hausdorff spaces, in addition to satisfying the usual requirements.) Kadison established such a result in (6, Theorem 6.6) for any Banach algebra with unit, if the algebra satisfies an appropriate reality condition and has a norm N such that N(x)2 — N(x2) for all x.

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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