On 𝜇-spaces and 𝑘_{𝑅}-spaces

Blasco, Jose L.

doi:10.1090/s0002-9939-1977-0464152-7

Cited by 8 publications

(8 citation statements)

References 19 publications

(16 reference statements)

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“…A topological space X is called µ-complete if each bounded subset of X has compact closure, where a subset B ⊂ X is bounded if for any locally finite collection U of open sets in X only finitely many sets U ∈ U meet B, see [15]. It is easily seen (and well-known) that a subset B of a Tychonoff space is bounded if and only if for any continuous real-valued function f : X → R the image f (B) is bounded in R. According to [31, 8.5.13], each Dieudonnécomplete space (in particular, each paracompact space) is µ-complete.…”

Section: Subproper Mapsmentioning

confidence: 99%

k*-Metrizable spaces and their applications

2008

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In this paper we introduce and study so-called k * -metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory.By definition, a Hausdorff topological space X is k * -metrizable if X is the image of a metrizable space M under a continuous map f : M → X having a section s : X → M that preserves precompact sets in the sense that the image s(K) of any compact set K ⊂ X has compact closure in X.Contents 42 13. The structure of sequential k * -metrizable groups 47 References 49 1 Proposition 1.3. Let X, Y be topological spaces such that X is µ-complete and each compact subset of Y is sequentially compact. A function s : Y → X is precompact-preserving if and only if s is cs * -continuous.Proof. The "only if" part is trivial and holds without any assumptions on X and Y . To prove the "if" part, assume that s : Y → X is a cs * -continuous function from a space Y whose all compact subsets are sequentially compact into a µ-complete space X. To show that s is precompact-preserving, take any compact subset K ⊂ Y . We claim that the image s(K) is precompact. Assuming that it is not so, we get that s(K) is not bounded by the µ-completeness of X. Consequently there is an infinite locally finite family U = {U n : n ∈ ω}

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Section: Subproper Mapsmentioning

confidence: 99%

k*-Metrizable spaces and their applications

2008

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“…Recall that a subset B of X is bounded in X iff[B] is a bounded subset of [w for e a c h f e C ( X ) : this concept was introduced in [ I ] and [ 3 ] , independently. It is not hard to see that a subset B of a space X is bounded in X iff for every sequence (V,,),,,, of nonempty open subsets of X with B n V,, # 0, for all iz < (0, there are x E X and p E o* such that x E U p , (V,,),,,,,).…”

Section: P-pseudocompactnessmentioning

confidence: 99%

“…( 2 ) 3 (3). Let (V,l),li(o be a sequence of nonempty open subsets of T(9) and let {A,, : IZ < 0 ) be an (11-partition o f (I) such that 9 E L ( p , (A,,*),,<J.…”

Section: Proofmentioning

confidence: 99%

Some Generalizations of Pseudocompactness

Garcı́a-Ferreira

1994

Annals of the New York Academy of Sciences

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In this paper, we introduce the concepts of p‐boundedness for pɛω*, (α, M)‐pseudocompactness and (α, M)‐compactness, for a cardinal number α and Ø≠M⊆β(ω)\ω. We prove that Xα is pseudocompact (respectively, countably compact) iff X is (α, M)‐pseudocompact (respectively, (α, M)‐compact), for some Ø≠M⊆β(ω)\ω; the Rudin‐Keisler order on β(ω)\ω can be defined in terms of p‐boundedness and p‐pseudocompactness; and if pɛβ(ω)\ω then p is RK‐minimal (selective) iff the space ω∪T(p) is p‐pseudocompact, where T(p) is the type of p in β(ω)\ω.

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“…Then pT is the smallest subspace of ßT which contains T and has the property that its bounded subsets have compact closure. Buchwalter [11] gives a construction of pT; see also [8], [10], [30], [34]. F is said to be a p-space if F = jtiF.…”

Section: Basic Definitions and Terminology Throughout F Denotes A Cmentioning

confidence: 99%

Weak and pointwise compactness in the space of bounded continuous functions

Wheeler¹

1981

Trans. Amer. Math. Soc.

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Abstract.Let T be a completely regular Hausdorff space, Cb(T) the space of bounded continuous real-valued functions on T, M(T) the Banach space dual of Cb(T). Let % denote the family of subsets of Cb(T) which are uniformly bounded and relatively compact for the topology 9^ of pointwise convergence. A well-known theorem of Grothendieck [28] asserts that if F is a compact Hausdorff space, and H is a uniformly bounded, pointwise compact set of real-valued continuous functions on F, then H is compact in the weak topology of the Banach space CÍT). Thus, in this case, the weak and pointwise topologies agree on//. This article is concerned with extensions of Grothendieck's Theorem to the setting: F is a completely regular Hausdorff space, and CbiT) is the Banach space of bounded continuous real-valued functions on T, with the supremum norm. Ptak [51] and Tomasek [64] have shown that Grothendieck's Theorem as stated above holds if and only if T is pseudocompact. However, one can pursue the matter as follows:Main Problem. For a given completely regular T, what is the largest linear subspace Z of MÎT), the Banach space dual of CbiT), such that the pointwise topology and the weak topology

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On 𝜇-spaces and 𝑘_{𝑅}-spaces

Cited by 8 publications

References 19 publications

k*-Metrizable spaces and their applications

k*-Metrizable spaces and their applications

Some Generalizations of Pseudocompactness

Weak and pointwise compactness in the space of bounded continuous functions

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