Compact and extremally disconnected spaces
Viglino defined a Hausdorff topological space to be C-compact if each closed subset of the space is an H-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is an S-set in the sense of Dickman and Krystock. Such spa...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2004-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171204208249 |
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| Summary: | Viglino defined a Hausdorff topological space to be C-compact
if each closed subset of the space is an H-set in the sense of
Veličko. In this paper, we study the class of Hausdorff spaces
characterized by the property that each closed subset is an
S-set in the sense of Dickman and Krystock. Such spaces are
called C-s-compact. Recently, the notion of strongly
subclosed relation, introduced by Joseph, has been utilized to
characterize C-compact spaces as those with the property that
each function from the space to a Hausdorff space with a strongly
subclosed inverse is closed. Here, it is shown that
C-s-compact spaces are characterized by the property that
each function from the space to a Hausdorff space with a strongly
sub-semiclosed inverse is a closed function. It is established
that this class of spaces is the same as the class of Hausdorff,
compact, and extremally disconnected spaces. The class of
C-s-compact spaces is properly contained in the class of
C-compact spaces as well as in the class of S-closed spaces
of Thompson. In general, a compact space need not be
C-s-compact. The product of two C-s-compact spaces need
not be C-s-compact. |
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| ISSN: | 0161-1712 1687-0425 |