Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Baire measure; realcompactness; local realcompactness; HN-completeness
Baire measure; realcompactness; local realcompactness; HN-completeness
Summary:
Two classes of spaces are studied, namely locally realcompact spaces and HN-complete spaces, where the latter class is introduced in the paper. Both of these classes are superclasses of the class of realcompact spaces. Invariance with respect to subspaces and products of these spaces are investigated. It is shown that these two classes can be characterized by demanding that certain equivalences hold between certain classes of Baire measures or by demanding that certain classes of Baire measures have non empty support. It is known that a space is locally realcompact if and only if it is open in its Hewitt-Nachbin realcompactification; we give an external characterization of HN-completeness with respect to the Hewitt-Nachbin realcompactification. In addition, a complete characterization of products of these classes is given.
References:
[1] Arhangelskiĭ A.V.: On topological spaces which are complete in the sense of Čech. Vestnik Moskov. Univ. Ser. I Mat. Meh. (1961), 2 37-40 (Russian). MR 0131258
[2] Blair R.L., van Douwen E.: Nearly realcompact spaces. Topology Appl. 47 3 (1992), 209-221. MR 1192310 | Zbl 0772.54021
[3] Buhagiar D., Chetcuti E., Dvurečenskij A.: Measure-theoretic characterizations of certain topological properties. Bull. Pol. Acad. Sci. 53 1 (2005), 99-109. MR 2162757 | Zbl 1113.28012
[4] Čech E.: On bicompact spaces. Ann. Math. 38 (1937), 823-844. MR 1503374
[5] Dijkstra J.J.: Measures in topology. Master Thesis, Univ. of Amsterdam, 1977.
[6] Dykes N.: Generalizations of realcompact spaces. Pacific J. Math. 33 (1970), 571-581. MR 0276928 | Zbl 0197.19201
[7] Engelking R.: General Topology. Heldermann, Berlin, 1989 (revised edition). MR 1039321 | Zbl 0684.54001
[8] Frolík Z.: Generalizations of the $G_\delta$-property of complete metric spaces. Czechoslovak Math. J. 10 (1960), 359-379. MR 0116305
[9] Frolík Z.: A generalization of realcompact spaces. Czechoslovak Math. J. 13 (1963), 127-138. MR 0155289
[10] Gardner R.J., Pfeffer W.F.: Borel Measures. in: Handbook of Set-Theoretic Topology, Elsevier, 1984, pp.961-1043. MR 0776641 | Zbl 0593.28016
[11] Gillman L., Jerison M.: Rings of Continuous Functions. Springer, New York, 1976. MR 0407579 | Zbl 0327.46040
[12] Hart K.P., Nagata J., Vaughan J.E. (Eds.): Encyclopedia of General Topology. Elsevier Science Publishers B.V., Amsterdam, The Netherlands, 2004. MR 2049453 | Zbl 1059.54001
[13] Hewitt E.: Linear functionals on spaces of continuous functions. Fund. Math. 37 (1950), 161-189. MR 0042684 | Zbl 0040.06401
[14] Isiwata T.: On locally Q-complete spaces, I, II, and III. Proc. Japan Acad. 35 (1959), 232-236, 263-267, 431-434. MR 0107220
[15] Mack J., Rayburn M., Woods G.: Lattices of topological extensions. Trans. Amer. Math. Soc. 189 (1972), 163-174. MR 0350700
[16] Nagata J.: Modern General Topology. Elsevier Science Publishers B.V., Amsterdam, The Netherlands, 1985, Second revised edition. Zbl 0598.54001
[17] Rice M.D., Reynolds G.D.: Weakly Borel-complete topological spaces. Fund. Math. 105 (1980), 179-185. MR 0580580 | Zbl 0435.54033
[18] Sakai M.: A new class of isocompact spaces and related results. Pacific J. Math. 122 1 (1986), 211-221. MR 0825232 | Zbl 0592.54025
[19] Weir M.: Hewitt-Nachbin Spaces. North Holland Math. Studies, American Elsevier, New York, 1975. MR 0514909 | Zbl 0314.54002
Search
Partner of
