Merotopological spaces
type: Item
instance of: scholarly article
title: Merotopological spaces
zbMATH Open document ID: 1052.54022
DOI: 10.1023/B:APCS.0000018143.05864.92
author: H. Lamar Bentley
author: Horst Herrlich
published in: Applied Categorical Structures
publication date: 2004-05-27T00:00:00Z
review text: A triple \(X = (X, \tau,\xi)\) is called a merotopological space if it satisfies the following axioms. \((M1)\) \((X,\tau)\) is a topological space; (M2) \((X, \xi)\) is a merotopic space in the sense of \textit{M. Katětov}, where \(\xi\) is the collection consisting of all collections of subsets of \(X\) which are near [Proc. Int. Congr. Math. 1962, 473--479 (1963; Zbl 0116.14105)]; (M3) whenever \(\{\text{cl}_\tau A \mid A \in {\mathcal A}\}\in \xi\), then \(\{A\mid A\in {\mathcal A}\}\in\xi\). It is shown that by defining morphisms in a natural way a topological category \textbf{MerTop} is obtained which contains the categories \textbf{Top} of (all!) topological spaces and \textbf{Near} of nearness spaces as fully embedded subcategories. \textbf{Near} is bireflective in \textbf{MerTop}. Whereas it is still an open problem to characterize internally the spaces belonging to the epireflective hull of the fully embedded subcategory \textbf{TopS} of symmetric \((= R_0)\) topological spaces in \textbf{Near}, it is shown that the the epireflective hull of \textbf{Top} in \textbf{MerTop} is identical with \textbf{MerTop}. The main tool for proving this interesting theorem is the construction of a so-called bunch completion \(X^+\) for each merotopological space \(X\). It turns out that, for regular nearness spaces \(X\), the strict completion \(X^*\) constructed by \textit{H. Herrlich} in [General Topology Appl. 4, 191--212 (1974; Zbl 0288.54004)] is a retract of \(X^+\). Moreover, it is shown that every topological space \((X,\tau)\) can be turned in a natural way into a merotopological space \(X = (X,\tau,\xi)\) such that the Wyler prime closed filter compactification \(\lambda X\) of \((X,\tau)\) is isomorphic to a subspace of \(X^+\). Finally, a merotopological version of the classical Taĭmanov extension theorem is proved which contains the original extension theorem for topological spaces (without any restriction!) as a special case.
Mathematics Subject Classification ID: 54E17
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54D80
Mathematics Subject Classification ID: 54C20
zbMATH DE Number: 2069147
zbMATH Keywords: merotopological space
zbMATH Keywords: nearness space
zbMATH Keywords: topological category
zbMATH Keywords: epireflective hull
zbMATH Keywords: bunch completion
zbMATH Keywords: strict completion
zbMATH Keywords: Wyler prime closed filter compactification
zbMATH Keywords: extension of mappings
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
Recommended article: Approach merotopological spaces and their completion
Recommended article: Q5440736
Recommended article: Q4825696
Recommended article: Mereotopological connection
Recommended article: Equilogical spaces
Recommended article: On Heterogeneous Spaces
Recommended article: Interpreting mereotopological connection
Recommended article: Neometric spaces
Recommended article: Bornological spaces
description: scientific article; zbMATH DE number 2069147