Geodesic
Some properties of pre-uniform spaces
Filomat, Tome 37 (2023) no. 10, p. 3201

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DOI

In this paper, we introduce the notions of pre-uniform spaces and pre-proximities and investigate some basic properties about them, where the definition of pre-uniformity here is different from the pre-uniformities which are studied in [1], [8] and [12], respectively. First, we prove that each pre-uniform pre-topology is regular, and give an example to show that there exists a pre-uniform structure on a finite set such that the pre-uniform pre-topology is not discrete. Moreover, we give three methods of generating (strongly) pre-uniformities, that is, the definition of a pre-base, a family of strongly pre-uniform covers, or a family of strongly pre-uniform pseudometrics. As an application, we show that each strongly pre-topological group is completely regular. Finally, we pose the concept of the pre-proximity on a set and discuss some properties of the pre-proximity.
DOI : 10.2298/FIL2310201L
Classification : 13A99 54C08, 54E05;54E15
Keywords: Pre-topological space, pre-uniform space, pre-uniformity, pre-proximity, strongly pre-uniform, almost uniform structure, symmetrically pre-uniform 
Fucai Lin; Yufan Xie; Ting Wu; Meng Bao. Some properties of pre-uniform spaces. Filomat, Tome 37 (2023) no. 10, p. 3201 . doi: 10.2298/FIL2310201L
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     title = {Some properties of pre-uniform spaces},
     journal = {Filomat},
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     year = {2023},
     volume = {37},
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     doi = {10.2298/FIL2310201L},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2310201L/}
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