Distinguished $$ C_{p}(X) $$ spaces

Ferrando, J. C.; Ka̧kol, Jerzy; Leiderman, Arkady; Saxon, Stephen A.

doi:10.1007/s13398-020-00967-4

Cited by 17 publications

(27 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A family {N x : x ∈ X} of subsets of a Tychonoff space X is called a scant cover for X if each N x is an open neighbourhood of x and for each u ∈ X the set X u = {x ∈ X : u ∈ N x } is finite. 1 Our Theorem 2.1 generalizes one of the results obtained in [14] stating that if X admits a scant cover…”

Section: Corollary 22 ([14]supporting

confidence: 67%

“…A very natural question arises about what those scattered compact spaces X ∈ Δ are. In view of Theorem 2.1, it is known that a Corson compact X belongs to the class Δ if and only if X is a scattered Eberlein compact space [14]. With the help of Theorems 2.1 and 3.8 we show that the class Δ contains also all separable compact spaces of the Isbell-Mrówka type.…”

Section: Definition 12 ([21])mentioning

confidence: 84%

“…Remark 2.4. The following applicable concept has been re-introduced in [14]. A family {N x : x ∈ X} of subsets of a Tychonoff space X is called a scant cover for X if each N x is an open neighbourhood of x and for each u ∈ X the set X u = {x ∈ X : u ∈ N x } is finite.…”

Section: Corollary 22 ([14]mentioning

confidence: 99%

“…For spaces C p (X) we proved in [14] the following theorem (the equivalence (1) ⇔ (4) has been obtained in [12]). Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…Several examples of C p (X) with(out) distinguished property have been provided in papers [12], [13] and [14]. The aim of this research is to continue our initial work on distinguished spaces C p (X).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications

Ka̧kol¹,

Leiderman²

2021

Proc. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

We prove that the locally convex space C p (X) of continuous realvalued functions on a Tychonoff space X equipped with the topology of pointwise convergence is distinguished if and only if X is a Δ-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45-60]. As an application of this characterization theorem we obtain the following results: 1) If X is aČech-complete (in particular, compact) space such that C p (X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell-Mrówka type X, the space C p (X) is distinguished. 3) If X is the compact space of ordinals [0, ω 1 ], then C p (X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that C p (X) is distinguished, is independent of ZFC. We also explore the question to which extent the class of Δ-spaces is invariant under basic topological operations.

show abstract

Section: Corollary 22 ([14]supporting

confidence: 67%

Section: Definition 12 ([21])mentioning

confidence: 84%

Section: Corollary 22 ([14]mentioning

confidence: 99%

“…For spaces C p (X) we proved in [14] the following theorem (the equivalence (1) ⇔ (4) has been obtained in [12]). Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%