Previous |  Up |  Next

Article

Keywords:
Hausdorff (Tychonoff) mapping; compactification of a mapping; maximal Hausdorff (Tychonoff) compactification of a mapping; perfect compactification of a mapping
Summary:
We prove that the maximal Hausdorff compactification $\chi f$ of a $T_2$-compactifi\-able mapping $f$ and the maximal Tychonoff compactification $\beta f$ of a Tychonoff mapping $f$ (see [P]) are perfect. This allows us to give a characterization of all perfect Hausdorff (respectively, all perfect Tychonoff) compactifications of a $T_2$-compactifiable (respectively, of a Tychonoff) mapping, which is a generalization of two results of Skljarenko [S] for the Hausdorff compactifications of Tychonoff spaces.
References:
[A] Arhangel'skii A.V.: Relative topological properties and relative topological spaces. Topology Appl. 70 (1996), 87-99. MR 1397067 | Zbl 0848.54016
[BN] Bludova I.V., Nordo G.: On the posets of all the Hausdorff and all the Tychonoff compactifications of continuous mappings. Questions & Answers Gen. Topology 17 (1999), 47-55. MR 1683951
[C] Chaber J.: Remarks on open-closed mappings. Fund. Math. 74 (1972), 197-208. MR 0303487 | Zbl 0242.54009
[D] Diamond B.: A characterization of those spaces having zero-dimensional remainders. Rocky Mountain J. Math. 15 1 (1985), 47-60. MR 0779251 | Zbl 0572.54022
[E] Engelking R.: General Topology. Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[HI] Henriksen M., Isbell J.R.: Some properties of compactifications. Duke Math. J. 25 (1958), 83-106. MR 0096196 | Zbl 0081.38604
[M] Mazroa E.M.R.: Perfect compactifications of continuous mappings. Vestn. Mosk. Univ. Ser. I (1990), no. 1, 23-26; Moscow Univ. Math. Bull. 45 (1990), no. 1, 24-26. MR 1064275
[N] Nordo G.: A basic approach to the perfect extensions of spaces. Comment. Math. Univ. Carolinae 38.3 (1997), 571-580. MR 1485078 | Zbl 0937.54014
[No] Norin V.P.: On proximities for mappings. Vestn. Mosk. Univ. Math. Mech. Ser. I 1982, no. 4 (1982), 33-36; Moscow Univ. Math. Bull. 37, no. 4 (1982), 40-44. MR 0671884 | Zbl 0508.54023
[P] Pasynkov B.A.: On extension to mappings of certain notions and assertions concerning spaces. in: Mapping and Functors, Izdat. MGU, Moscow, 1984, pp.72-102 (in Russian). MR 0791256
[S] Skljarenko E.G.: On perfect bicompact extensions. Dokl. Akad. Nauk SSSR 137 (1961), 39-41; English transl.: Soviet Math. Dokl. 2 (1961), 238-240. MR 0121777
[W] Whyburn G.T.: Compactification of mappings. Math. Ann. 166 (1966), 168-174. MR 0200905 | Zbl 0141.20505
Partner of
EuDML logo