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

  • Received : 2009.05.19
  • Accepted : 2009.06.09
  • Published : 2009.06.25

Abstract

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).

Keywords

function space;Fr$\'{e}$chet-Urysohn;AP;${\tau}$-monolithic;strongly ${\tau}$-monolithic;countable fan-tightness;Lindel$\"{o}$f

References

  1. A. V. Arhangel'skii, Some topological spaces that arise in functional analysis Uspekhi Mat. Nauk 31(5) (1976) 17-32. MR 55 #16569, Russian Math. Surveys 31(5) (1976), 14-30.
  2. A. V. Arhangel'skii, Factorization theorems and function spaces: stability and monolithicity, Soviet Math. Dokl. 26 (1982), 177-181.
  3. A. V. Arhangel'skii. Hurewicz spaces, analytic sets and fan-tightness of spaces of functions, Soviet Math. Dokl. 33 (1986), 396-399.
  4. A. V. Arhangel'skii, Topological function spaces, Kluwer Academic Publishers, 1992.
  5. A. Bella and I. V. Yaschenko, On AP and WAP spaces, Comment. Math. Univ. Carolinae, 40(3) (1999), 531-536.
  6. J. Cao, J. Kim, T. Nogura, and Y. Song, Cardinal invariants related to star covering properties, Topology Proc., 26(1) (2001-2002), 83-96.
  7. R. A. McCoy, K-space function spaces, Internat. J. Math. & Math. Sci., 3(4) (1980), 701-711. https://doi.org/10.1155/S0161171280000506
  8. V.V. Tkachuk and I.V. Yaschenko, Almost closed sets and topologies they determine, Comment. Math. Univ. Carolinae, 42(2) (2001), 393-403.