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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 University^{a} 
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. 277283. 
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