Displaying 1541 – 1560 of 1977

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Some properties of relatively strong pseudocompactness

Guo-Fang Zhang (2008)

Czechoslovak Mathematical Journal

In this paper, we study some properties of relatively strong pseudocompactness and mainly show that if a Tychonoff space X and a subspace Y satisfy that Y Int Y ¯ and Y is strongly pseudocompact and metacompact in X , then Y is compact in X . We also give an example to demonstrate that the condition Y Int Y ¯ can not be omitted.

Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane

Adam St. Arnaud, Piotr Rudnicki (2013)

Formalized Mathematics

We first provide a modified version of the proof in [3] that the Sorgenfrey line is T1. Here, we prove that it is in fact T2, a stronger result. Next, we prove that all subspaces of ℝ1 (that is the real line with the usual topology) are Lindel¨of. We utilize this result in the proof that the Sorgenfrey line is Lindel¨of, which is based on the proof found in [8]. Next, we construct the Sorgenfrey plane, as the product topology of the Sorgenfrey line and itself. We prove that the Sorgenfrey plane...

Some remarks on the product of two C α -compact subsets

Salvador García-Ferreira, Manuel Sanchis, Stephen W. Watson (2000)

Czechoslovak Mathematical Journal

For a cardinal α , we say that a subset B of a space X is C α -compact in X if for every continuous function f X α , f [ B ] is a compact subset of α . If B is a C -compact subset of a space X , then ρ ( B , X ) denotes the degree of C α -compactness of B in X . A space X is called α -pseudocompact if X is C α -compact into itself. For each cardinal α , we give an example of an α -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness”...

Some results and problems about weakly pseudocompact spaces

Oleg Okunev, Angel Tamariz-Mascarúa (2000)

Commentationes Mathematicae Universitatis Carolinae

A space X is truly weakly pseudocompact if X is either weakly pseudocompact or Lindelöf locally compact. We prove: (1) every locally weakly pseudocompact space is truly weakly pseudocompact if it is either a generalized linearly ordered space, or a proto-metrizable zero-dimensional space with χ ( x , X ) > ω for every x X ; (2) every locally bounded space is truly weakly pseudocompact; (3) for ω < κ < α , the κ -Lindelöfication of a discrete space of cardinality α is weakly pseudocompact if κ = κ ω .

Some results on L Σ ( κ ) -spaces

Fidel Casarrubias Segura, Oleg Okunev, Paniagua C. G. Ramírez (2008)

Commentationes Mathematicae Universitatis Carolinae

We present several results related to L Σ ( κ ) -spaces where κ is a finite cardinal or ω ; we consider products and some constructions that lead from spaces of these classes to other spaces of similar classes.

Some results on semi-stratifiable spaces

Wei-Feng Xuan, Yan-Kui Song (2019)

Mathematica Bohemica

We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If X is a semi-stratifiable space, then X is separable if and only if X is D C ( ω 1 ) ; (2) If X is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then X is separable; (3) Let X be a ω -monolithic star countable extent semi-stratifiable space. If t ( X ) = ω and d ( X ) ω 1 , then X is hereditarily separable. Finally, we prove that for any T 1 -space...

Some results on sequentially compact extensions

Maria Cristina Vipera (1998)

Commentationes Mathematicae Universitatis Carolinae

The class of Hausdorff spaces (or of Tychonoff spaces) which admit a Hausdorff (respectively Tychonoff) sequentially compact extension has not been characterized. We give some new conditions, in particular, we prove that every Tychonoff locally sequentially compact space has a Tychonoff one-point sequentially compact extension. We also give some results about extension of functions and about lattice properties of the family of all minimal sequentially compact extensions of a given space.

Currently displaying 1541 – 1560 of 1977