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Russia Izhevsk
Section Mathematics
Title On convergent sequences and properties of subspaces
Author(-s) Gryzlov A.A.a, Tsigvintseva K.N.a
Affiliations Udmurt State Universitya
Abstract We consider problems connected with the notion of convergent sequences in $T_1$-spaces. The properties of $T_1$-spaces and convergent sequences in these spaces considerably differ from the same properties of Hausdorff spaces. We consider problems connected with the properties of the minimal $T_1$-space. We consider properties of spaces where every sequence is a convergent sequence (Theorems 1 and 2 and Example 1). One of the main problems is the connection between convergent sequences and the properties of subspaces of the space. It is well known that the compactness, countable compactness and sequential compactness are not equivalent in general. We prove (Theorem 7) that hereditary sequential compactness, compactness and countable compactness are equivalent.
Keywords convergent sequence, $T_1$-compactness, compactness
UDC 515.122.22, 515.122.252
MSC 54D10, 54D30
DOI 10.20537/vm180301
Received 25 July 2018
Language Russian
Citation Gryzlov A.A., Tsigvintseva K.N. On convergent sequences and properties of subspaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 3, pp. 277-283.
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