# Article

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Keywords:
tightness; fan-tightness; countably compact spaces; pseudo-compact space; P-point; biquotient mapping
Summary:
Countable tightness is compared to the stronger notion of countable fan-tight\-ness. In particular, we prove that countable tightness is equivalent to countable fan-tightness in countably compact regular spaces, and that countable fan-tightness is preserved by pseudo-open compact mappings. We also discuss the behaviour of countable tightness and of countable fan-tightness under the product operation.
References:
[1] Arens R.: Note on convergence in topology. Math. Mag. 23 (1950), 229-234. MR 0037500 | Zbl 0041.31502
[2] Arhangel'skii A.V.: Mappings and spaces. Russian Math. Surveys 21 (1966), 115-162. MR 0227950
[3] Arhangel'skii A.V.: Hurewicz spaces, analytic sets and fan-tightness of spaces of functions. Soviet Math. Dokl. 33 \number 2 (1986), 396-399.
[4] Arhangel'skii A.V.: The frequency spectrum of a topological space and the product operation. Trans. Moscow Math. Soc. 40 (1981), 163-200.
[5] Gruenhage G., Tanaka Y.: Products of \$k\$-spaces and spaces of countable tightness. Trans. Amer. Math. Soc. 273 \number 1 (1982), 299-308. MR 0664043 | Zbl 0491.54019
[6] Harley P.W. III, Stephenson R.M., Jr.: Symmetrizable and related spaces. Trans. Amer. Math. Soc. 219 (1976), 89-111. MR 0418048
[7] Kannan V.: Every compact \$T_5\$ sequential space is Fréchet. Fund. Math. 107 (1980), 85-90. MR 0584661 | Zbl 0452.54019
[8] Malykhin V.I.: On countable spaces having no bicompactification of countable tightness. Soviet Math. Dokl. 13 \number 5 (1972), 1407-1411.
[9] Malykhin V.I.: An example of a topological group. in Topological spaces and their mappings, Riga. Latv. Gosud. Univ. (in Russian), 1981, pp. 120-123. MR 0630428 | Zbl 0478.22001
[10] Michael E.A.: A quintuple quotient quest. Gen. Top. Appl. 2 (1972), 91-138. MR 0309045 | Zbl 0238.54009
[11] Nedev S.: Symmetrizable spaces and final compactness. Soviet Math. Dokl. 8 (1967), 890-892. MR 0216460 | Zbl 0153.52701
[12] Nyikos P.J.: Metrizability and the Fréchet-Urysohn property in topological groups. Proc. Amer. Math. Soc. 83 (1981), 793-801. MR 0630057 | Zbl 0474.22001
[13] Siwiec F.: Sequence-covering and countably biquotient mappings. Gen. Top. Appl. 1 (1971), 143-154. MR 0288737
[14] Siwiec F.: Generalizations of the first axiom of countability. Rocky Mount. J. Math. 5 (1975), 1-60. MR 0358699 | Zbl 0294.54021
[15] Siwiec F., Mancuso V.J.: Relations among certain mappings and conditions for their equivalence. Gen. Top. Appl. 1 (1971), 34-41. MR 0282347 | Zbl 0216.44203
[16] Stephenson M., Jr: Symmetrizable, \${\Cal F}\$-, and weakly first countable spaces. Can. J. Math. XXIX (1977), 480-488. MR 0442885
[17] Tamano K.: Closed images of metric spaces and metrization. Topology Proceedings 10 (1985), 177-186. MR 0851211 | Zbl 0616.54026
[18] Tanaka Y.: Product of spaces of countable tightness. Topology Proceedings 6 (1981), 115-133. MR 0650484
[19] Tanaka Y.: Metrizability of certain quotient spaces. Fund. Math. 119 (1983), 157-168. MR 0731817
[20] Todorcevic S.: Some applications of S and L combinatorics. Annals New York Academy of Science, 1991, pp. 130-167. MR 1277886 | Zbl 0836.54018
[21] Uspenskii V.V.: Frequency spectrum of functional spaces. Vestnik Mosk. Universiteta, Ser. Matematica 37\number 1 (1982), 31-35.

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