Metrization Theorem for Uniform Loops with the Invertibility Property

Dagmar Markechová; Peter Vrábel; Beáta Stehlíková

doi:10.3390/math5020032

On the metrizability of TVS-cone metric spaces

Publications de l Institut Mathematique ◽

10.2298/pim1512271l ◽

2015 ◽

Vol 98 (112) ◽

pp. 271-279 ◽

Cited By ~ 2

Author(s):

Shou Lin ◽

Kedian Li ◽

Ying Ge

Keyword(s):

Compact Space ◽

Metric Spaces ◽

Cone Metric Spaces ◽

Metric Topology ◽

Cone Metric ◽

Metrization Theorem

Metric spaces are cone metric spaces, and cone metric spaces are TVS-cone metric spaces. We prove that TVS-cone metric spaces are paracompact. A metrization theorem of TVS-cone metric spaces is obtained by a purely topological tools. We obtain that a homeomorphism f of a compact space is expansive if and only if f is TVS-cone expansive. In the end, for a TVS-cone metric topology, a concrete metric generating the topology is constructed.

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Separating Closed Sets by Continuous Mappings into Developable Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-108-1 ◽

1981 ◽

Vol 33 (6) ◽

pp. 1420-1431 ◽

Cited By ~ 7

Author(s):

Harald Brandenburg

Keyword(s):

Topological Space ◽

Double Sequence ◽

Completely Regular ◽

Closed Sets ◽

Continuous Mappings ◽

Neighbourhood Base ◽

Image Position ◽

Metrizable Spaces ◽

Metrization Theorem ◽

Developable Spaces

A topological space X is called developable if it has a development, i.e., a sequence of open covers of X such that for each x ∈ X the collection is a neighbourhood base of x, whereThis class of spaces has turned out to be one of the most natural and useful generalizations of metrizable spaces [23]. In [4] it was shown that some well known results in metrization theory have counterparts in the theory of developable spaces (i.e., Urysohn's metrization theorem, the Nagata-Smirnov theorem, and Nagata's “double sequence theorem”). Moreover, in [3] it was pointed out that subspaces of products of developable spaces (i.e., D-completely regular spaces) can be characterized in much the same way as subspaces of products of metrizable spaces (i.e., completely regular T1-spaces).

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Essentials of General Topology

Fundamentals of Mathematical Analysis ◽

10.1093/oso/9780198868781.003.0005 ◽

2021 ◽

pp. 191-244

Author(s):

Adel N. Boules

Keyword(s):

Topological Spaces ◽

Hausdorff Spaces ◽

Locally Compact ◽

Compact Spaces ◽

Section 8 ◽

Finite Products ◽

The One ◽

One Point Compactification ◽

Locally Compact Spaces ◽

Metrization Theorem

The first eight sections of this chapter constitute its core and are generally parallel to the leading sections of chapter 4. Most of the sections are brief and emphasize the nonmetric aspects of topology. Among the topics treated are normality, regularity, and second countability. The proof of Tychonoff’s theorem for finite products appears in section 8. The section on locally compact spaces is the transition between the core of the chapter and the more advanced sections on metrization, compactification, and the product of infinitely many spaces. The highlights include the one-point compactification, the Urysohn metrization theorem, and Tychonoff’s theorem. Little subsequent material is based on the last three sections. At various points in the book, it is explained how results stated for the metric case can be extended to topological spaces, especially locally compact Hausdorff spaces. Some such results are developed in the exercises.

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