On locally compact Hausdorff spaces with finite metrizability number

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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Spaces which Cannot be Written as a Countable Disjoint Union of Closed Subsets

Canadian Mathematical Bulletin ◽

10.4153/cmb-1973-069-1 ◽

1973 ◽

Vol 16 (3) ◽

pp. 435-437 ◽

Cited By ~ 1

Author(s):

C. Eberhart ◽

J. B. Fugate ◽

L. Mohler

Keyword(s):

Hausdorff Space ◽

Disjoint Union ◽

Hausdorff Spaces ◽

Locally Compact ◽

The One ◽

One Point Compactification ◽

Compact Subsets

It is well known (see [3](1)) that no continuum (i.e. compact, connected, Hausdorff space) can be written as a countable disjoint union of its (nonvoid) closed subsets. This result can be generalized in two ways into the setting of locally compact, connected, Hausdorff spaces. Using the one point compactification of a locally compact, connected, Hausdorff space X one can easily show that X cannot be written as a countable disjoint union of compact subsets. If one makes the further assumption that X is locally connected, then one can show that X cannot be written as a countable disjoint union of closed subsets.(2)

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Simple Links in Locally Compact Connected Hausdorff Spaces are Nondegenerate

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-022-2 ◽

1981 ◽

Vol 34 (2) ◽

pp. 349-355

Author(s):

David John

Keyword(s):

Hausdorff Space ◽

Metric Spaces ◽

Connected Space ◽

Hausdorff Spaces ◽

Locally Compact ◽

Peano Continua

The fact that simple links in locally compact connected metric spaces are nondegenerate was probably first established by C. Kuratowski and G. T. Whyburn in [2], where it is proved for Peano continua. J. L. Kelley in [3] established it for arbitrary metric continua, and A. D. Wallace extended the theorem to Hausdorff continua in [4]. In [1], B. Lehman proved this theorem for locally compact, locally connected Hausdorff spaces. We will show that the locally connected property is not necessary.A continuum is a compact connected Hausdorff space. For any two points a and b of a connected space M, E(a, b) denotes the set of all points of M which separate a from b in M. The interval ab of M is the set E(a, b) ∪ {a, b}.

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A Measure Theoretical Version of the Aleksandrov Theorem

Georgian Mathematical Journal ◽

10.1515/gmj.2008.53 ◽

2008 ◽

Vol 15 (1) ◽

pp. 53-61

Author(s):

Majid Gazor

Keyword(s):

Borel Measure ◽

Measure Theory ◽

Ordinal Number ◽

Hausdorff Spaces ◽

Locally Compact ◽

Closed Sets

Abstract In this paper a theorem analogous to the Aleksandrov theorem is presented in terms of measure theory. Furthermore, we introduce the condensation rank of Hausdorff spaces and prove that any ordinal number is associated with the condensation rank of an appropriate locally compact totally imperfect space. This space is equipped with a probability Borel measure which is outer regular, vanishes at singletons, and is also inner regular in the sense of closed sets.

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A Compactification due to Fell

Canadian Mathematical Bulletin ◽

10.4153/cmb-1972-028-3 ◽

1972 ◽

Vol 15 (1) ◽

pp. 145-146 ◽

Cited By ~ 2

Author(s):

Aubrey Wulfsohn

Keyword(s):

Fundamental System ◽

Point Of View ◽

Hausdorff Spaces ◽

Locally Compact ◽

Alternative Construction ◽

Continuous Fields ◽

One Point Compactification

We give an alternative construction of a Hausdorff compactification due to Fell [2]. We say that a space is compact if it has the Heine-Borel property, locally compact if each point has a fundamental system of compact neighbourhoods. The interesting spaces from the point of view of this paper, are the non-Hausdorjf ones since for locally compact Hausdorff spaces Fell's compactification is the usual one-point compactification. The motivation for the compactification comes from the theory of continuous fields of C*-algebras: the primitive spectrum of a C*- algebra A is a locally compact T0 space X and Fell [3] realizes A as an algebra of fields of operators over the compactification of X. This note is based on a discussion of the author with Professor Fell.

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On Product of Radón Measures

Canadian Mathematical Bulletin ◽

10.4153/cmb-1969-053-x ◽

1969 ◽

Vol 12 (4) ◽

pp. 427-444 ◽

Cited By ~ 3

Author(s):

M. C. Godfrey ◽

M. Sion

Keyword(s):

Direct Reference ◽

Outer Measure ◽

Hausdorff Spaces ◽

Locally Compact ◽

Radon Measures ◽

Closed Sets

Let X, Y be locally compact Hausdorff spaces and μ, ν be Radón outer measures on X and Y respectively. The classical product outer measure ϕ on X × Y generated by measurable rectangles, without direct reference to the topology, turns out to have some serious drawbacks. For example, one can only prove that closed sets (and hence Baire sets) are ϕ-measurable. It is unknown, even when X and Y are compact, whether closed sets are ϕ-measurable.

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On projective lift and orbit spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013551 ◽

1994 ◽

Vol 50 (3) ◽

pp. 445-449 ◽

Cited By ~ 3

Author(s):

T.K. Das

Keyword(s):

Compact Hausdorff Space ◽

Hausdorff Space ◽

Covariant Functor ◽

Hausdorff Spaces ◽

Orbit Spaces ◽

Locally Compact ◽

Discontinuous Group ◽

Coreflective Subcategory ◽

Group Of Homeomorphisms ◽

Locally Compact Hausdorff Space

By constructing the projective lift of a dp-epimorphism, we find a covariant functor E from the category Cd of regular Hausdorff spaces and continuous dp-epimorphisms to its coreflective subcategory εd consisting of projective objects of Cd We use E to show that E(X/G) is homeomorphic to EX/G whenever G is a properly discontinuous group of homeomorphisms of a locally compact Hausdorff space X and X/G is an object of Cd.

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TOPOLOGICAL QUIVERS

International Journal of Mathematics ◽

10.1142/s0129167x05003077 ◽

2005 ◽

Vol 16 (07) ◽

pp. 693-755 ◽

Cited By ~ 10

Author(s):

PAUL S. MUHLY ◽

MARK TOMFORDE

Keyword(s):

Uniqueness Theorem ◽

Directed Graphs ◽

Algebra Structure ◽

Hausdorff Spaces ◽

Locally Compact ◽

Gauge Invariant ◽

C Algebras ◽

Natural Analogues ◽

General Perspective ◽

The Ideal

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver [Formula: see text] is a C*-correspondence, and from this correspondence one may construct a Cuntz–Pimsner algebra [Formula: see text]. In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with that theory much of the operator algebra structure of [Formula: see text] can be determined from [Formula: see text]. We also show that many fundamental results from the theory of graph C*-algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the gauge-invariant uniqueness theorem, the Cuntz–Krieger uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity.

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