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On half-completion and bicompletion of quasi-metric spaces

Elena Alemany, Salvador Romaguera (1996)

Commentationes Mathematicae Universitatis Carolinae

We characterize the quasi-metric spaces which have a quasi-metric half-completion and deduce that each paracompact co-stable quasi-metric space having a quasi-metric half-completion is metrizable. We also characterize the quasi-metric spaces whose bicompletion is quasi-metric and it is shown that the bicompletion of each quasi-metric compatible with a quasi-metrizable space X is quasi-metric if and only if X is finite.

On indecomposability and composants of chaotic continua

Hisao Kato (1996)

Fundamenta Mathematicae

A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x,y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that d ( f n ( x ) , f n ( y ) ) > c . A homeomorphism f: X → X is continuum-wise expansive if there is c > 0 such that if A is a nondegenerate subcontinuum of X, then there is an integer n ∈ ℤ such that d i a m i f n ( A ) > c . Clearly, every expansive homeomorphism is continuum-wise expansive, but the converse assertion is not true. In [6], we defined the notion of chaotic continua of homeomorphisms...

On subcompactness and countable subcompactness of metrizable spaces in ZF

Kyriakos Keremedis (2022)

Commentationes Mathematicae Universitatis Carolinae

We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space 𝐗 = ( X , T ) is countably compact if and only if it is countably subcompact relative to T . (iii) For every metrizable space 𝐗 = ( X , T ) , the following are equivalent: (a) 𝐗 is compact; (b) for every open filter of 𝐗 , { F ¯ : F } ; (c) 𝐗 is subcompact relative to T . We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable...

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