• Title/Summary/Keyword: countable $Fr\'{e}chet$-Urysohn

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ON $FR\'{E}CHET$-URYSOHN EXPANSIONS

  • Hong, Woo-Chorl
    • East Asian mathematical journal
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    • v.17 no.2
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    • pp.185-189
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    • 2001
  • In this paper we study on $Fr\'{e}chet$-Urysohn expansions of topological spaces, countable $Fr\'{e}chet$-Urysohn spaces and Hausdorff $Fr\'{e}chet$-Urysohn spaces.

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GENERALIZED FRÉCHET-URYSOHN SPACES

  • Hong, Woo-Chorl
    • Journal of the Korean Mathematical Society
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    • v.44 no.2
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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.

STRONG τ-MONOLITHICITY AND FRECHET-URYSOHN PROPERTIES ON Cp(X)

  • Kim, Jun-Hui;Cho, Myung-Hyun
    • Honam Mathematical Journal
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    • v.31 no.2
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    • pp.233-237
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    • 2009
  • In this paper, we show that: (1) every strongly ${\omega}$-monolithic space X with countable fan-tightness is Fr$\'{e}$chet-Urysohn; (2) a direct proof of that X is Lindel$\"{o}$f when $C_p$(X) is Fr$\'{e}$chet-Urysohn; and (3) X is Lindel$\"{o}$f when X is paraLindel$\"{o}$f and $C_p$(X) is AP. (3) is a generalization of the result of [8]. And we give two questions related to Fr$\'{e}$chet-Urysohn and AP properties on $C_p$(X).

A NOTE ON SPACES WHICH HAVE COUNTABLE TIGHTNESS

  • Hong, Woo-Chorl
    • Communications of the Korean Mathematical Society
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    • v.26 no.2
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    • pp.297-304
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    • 2011
  • In this paper, we introduce closure operators [${\cdot}$]c and [${\cdot}$]a on a space and study some relations among [${\cdot}$]c, [${\cdot}$]a and countable tightness. We introduce the concepts of a strongly sequentially closed set and a strongly sequentially open set and show that a space X has countable tightness if and only if every strongly sequentially closed set is closed if and only if every strongly sequentially open set is open. Finally we find a generalization of the weak Fr$\'{e}$chet-Urysohn property which is equivalent to countable tightness.

A NOTE ON SPACES DETERMINED BY CLOSURE-LIKE OPERATORS

  • Hong, Woo Chorl;Kwon, Seonhee
    • East Asian mathematical journal
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    • v.32 no.3
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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.

ON SPACES IN WHICH THE THREE MAIN KINDS OF COMPACTNESS ARE EQUIVALENT

  • Hong, Woo-Chorl
    • Communications of the Korean Mathematical Society
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    • v.25 no.3
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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 (*).