• 대한수학회지
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• 제44권2호
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• pp.261-273
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• 2007

In this paper, we introduce some new properties of a topological space which are respectively generalizations of $Fr\'{e}chet$-Urysohn property. We show that countably AP property is a sufficient condition for a space being countable tightness, sequential, weakly first countable and symmetrizable, to be ACP, $Fr\'{e}chet-Urysohn$, first countable and semimetrizable, respectively. We also prove that countable compactness is a sufficient condition for a countably AP space to be countably $Fr\'{e}chet-Urysohn$. We then show that a countably compact space satisfying one of the properties mentioned here is sequentially compact. And we show that a countably compact and countably AP space is maximal countably compact if and only if it is $Fr\'{e}chet-Urysohn$. We finally obtain a sufficient condition for the ACP closure operator $[{\cdot}]_{ACP}$ to be a Kuratowski topological closure operator and related results.

• East Asian mathematical journal
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• 제32권3호
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• pp.365-375
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• 2016

In this paper, we study some classes of spaces determined by closure-like operators $[{\cdot}]_s$, $[{\cdot}]_c$ and $[{\cdot}]_k$ etc. which are wider than the class of $Fr{\acute{e}}chet-Urysohn$ spaces or the class of sequential spaces and related spaces. We first introduce a WADS space which is a generalization of a sequential space. We show that X is a WADS and k-space iff X is sequential and every WADS space is C-closed and obtained that every WADS and countably compact space is sequential as a corollary. We also show that every WAP and countably compact space is countably sequential and obtain that every WACP and countably compact space is sequential as a corollary. And we show that every WAP and weakly k-space is countably sequential and obtain that X is a WACP and weakly k-space iff X is sequential as a corollary.

• 대한수학회논문집
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• 제22권2호
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• pp.297-303
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• 2007

In this paper, we study on C-closed spaces, SC-closed spaces and related spaces. We show that a sequentially compact SC-closed space is sequential and as corollaries obtain that a sequentially compact space with unique sequential limits is sequential if and only if it is C-closed [7, 1.19 Proposition] and every sequentially compact SC-closed space is C-closed. We also show that a countably compact WAP and C-closed space is sequential and obtain that a countably compact (or compact or sequentially compact) WAP-space with unique sequential limits is sequential if and only if it is C-closed as a corollary. Finally we prove that a weakly discretely generated AP-space is C-closed. We then obtain that every countably compact (or compact or sequentially compact) weakly discretely generated AP-space is $Fr\acute{e}chet$-Urysohn with unique sequential limits, for weakly discretely generated AP-spaces, unique sequential limits ${\equiv}KC{\equiv}C-closed{\equiv}SC-closed$, and every continuous surjective function from a countably compact (or compact or sequentially compact) space onto a weakly discretely generated AP-space is closed as corollaries.

• Korean Journal of Mathematics
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• 제29권3호
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• pp.555-561
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• 2021

In this paper, we investigate some properties of countably g-compact and sequentially GO-compact spaces. Also, we discuss the relation between countably g-compact and sequentially GO-compact. Next, we introduce the definition of g-subspace and study the characterization of g-subspace.

• 대한수학회보
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• 제41권2호
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• pp.235-239
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• 2004

Vaughan proved that if X is countably compact, then the Alexandroff duplicate A(X) is acc. In this note, we give an example to show that the result can not be generalized to star-Lindelof spaces. Moreover, we give a positive result.

• 대한수학회논문집
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• 제25권3호
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• pp.477-484
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• 2010

In this paper, we introduce a new property (*) of a topological space and prove that if X satisfies one of the following conditions (1) and (2), then compactness, countable compactness and sequential compactness are equivalent in X; (1) Each countably compact subspace of X with (*) is a sequential or AP space. (2) X is a sequential or AP space with (*).

• 호남수학학술지
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• 제36권2호
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• pp.425-434
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• 2014

In this paper, we introduce a new concept of a countably sequential space which is a generalization of a sequential space and study some properties of a countably sequential space and relations among the space and related spaces.

• 호남수학학술지
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• 제26권2호
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• pp.209-218
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• 2004

On the analogy of total countable compactness, we study interesting subfamilies in the class of pseudocompact spaces. We show relationships between totally pseudocompact spaces, sequentially pseudocompact spaces, and DFCC spaces. We also prove relationships among densely ${\xi}$-pseudocompact, ${\xi}$-pseudocompact, and countably pracompact spaces. As a productive result on countably pracompact spaces, we will prove that if X is a countably pracompact space and Y is a countably pracompact ${\kappa}$-space, then $X{\times}Y$ is count ably pracompact.

• 호남수학학술지
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• 제24권1호
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• pp.131-142
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• 2002

A $1{\frac{1}{2}}$-starcompact space has one of the most curious properties among the spaces of starcompactness. It is not too far away from countably compact spaces and may be considered as the first candidate for extending theorems about countably compact spaces. Unfortunately, $1{\frac{1}{2}}$-starcompactness is not so easy to be recognized as 2-starcompactness which will follow from countable pracompactness. We investigate some properties around $1{\frac{1}{2}}$-starcompact spaces.

• 충청수학회지
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• 제36권4호
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• pp.279-288
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• 2023

We obtain a Chandrabhan type fixed point theorem for a multimap having a non-compact domain and a weakly closed graph, and taking convex values only on a quasi-convex subset of Hausdorff locally convex topological vector space. We introduce the definition of Chandrabhan-set and find a sufficient condition for every countably condensing multimap to have a relatively compact Chandrabhan-set. Finally, we establish a new version of Sadovskii fixed point theorem for multimaps.