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Spaces with countable s n -networks

Ge Ying (2004)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we prove that a space X is a sequentially-quotient π -image of a metric space if and only if X has a point-star s n -network consisting of c s * -covers. By this result, we prove that a space X is a sequentially-quotient π -image of a separable metric space if and only if X has a countable s n -network, if and only if X is a sequentially-quotient compact image of a separable metric space; this answers a question raised by Shou Lin affirmatively. We also obtain some results on spaces with countable...

Spaces with property ( D C ( ω 1 ) )

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

We prove that if X is a first countable space with property ( D C ( ω 1 ) ) and with a G δ -diagonal then the cardinality of X is at most 𝔠 . We also show that if X is a first countable, DCCC, normal space then the extent of X is at most 𝔠 .

Spaces with σ -locally countable weak-bases

Zhaowen Li (2006)

Archivum Mathematicum

In this paper, spaces with σ -locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or g -first countable spaces with σ -locally countable c s -networks).

Stabilizers of closed sets in the Urysohn space

Julien Melleray (2006)

Fundamenta Mathematicae

Building on earlier work of Katětov, Uspenskij proved in [8] that the group of isometries of Urysohn's universal metric space 𝕌, endowed with the pointwise convergence topology, is a universal Polish group (i.e. it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group G, there exists a closed subset F of 𝕌 such that G is topologically isomorphic to the group of isometries of 𝕌 which map...

Statistical convergence of sequences of functions with values in semi-uniform spaces

Dimitrios N. Georgiou, Athanasios C. Megaritis, Selma Özçağ (2018)

Commentationes Mathematicae Universitatis Carolinae

We study several kinds of statistical convergence of sequences of functions with values in semi-uniform spaces. Particularly, we generalize to statistical convergence the classical results of C. Arzelà, Dini and P.S. Alexandroff, as well as their statistical versions studied in [Caserta A., Di Maio G., Kočinac L.D.R., {Statistical convergence in function spaces},. Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.] and [Caserta A., Kočinac L.D.R., {On statistical exhaustiveness}, Appl. Math. Lett....

Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets

Hisao Kato (1993)

Fundamenta Mathematicae

We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dim X > 0, then the decompositions W S ( x ) | x ∈ X and W ( u ) ( x ) | x ∈ X of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ϱ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, W σ ( x ) contains a nondegenerate...

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