Continuous and proper decompositions

Williams, G. K.

doi:10.1090/S0002-9939-1971-0273583-4

Continuous and proper decompositions
HTML articles powered by AMS MathViewer

by G. K. Williams PDF

Proc. Amer. Math. Soc. 28 (1971), 267-270 Request permission

Abstract:

If $X$ is a locally connected, locally peripherally compact Hausdorff space and if $R$ is an equivalence relation on $X$ with fibers which are connected with compact boundaries, then it is shown that three definitions of continuity of $R$ are equivalent. Some of the propositions used to obtain this result are then applied to get sufficient conditions for a decomposition of certain types of metric spaces to be proper.

References

Topologie

Henri Cartan, Quotients of complex analytic spaces, Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 1–15. MR 0139769
Jürgen Flachsmeyer, Über halbstetige Zerlegungen topologischer Räume, Math. Nachr. 24 (1962), 1–12 (German). MR 148024, DOI 10.1002/mana.19620240102
Harald Holmann, Komplexe Räume mit komplexen Transformations-gruppen, Math. Ann. 150 (1963), 327–360 (German). MR 150789, DOI 10.1007/BF01470762
Ingo Lieb, Über komplexe Räume and komplexe Spektren, Invent. Math. 1 (1966), 45–58 (German). MR 197776, DOI 10.1007/BF01389698
R. L. Moore, Foundations of point set theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR 0150722
Karl Stein, Analytische Zerlegungen komplexer Räume, Math. Ann. 132 (1956), 63–93 (German). MR 83045, DOI 10.1007/BF01343331
A. H. Stone, Metrizability of decomposition spaces, Proc. Amer. Math. Soc. 7 (1956), 690–700. MR 87078, DOI 10.1090/S0002-9939-1956-0087078-6
G. T. Whyburn, Continuous decompositions, Amer. J. Math. 71 (1949), 218–226. MR 27507, DOI 10.2307/2372107
G. T. Whyburn, Open mappings on locally compact spaces, Mem. Amer. Math. Soc. 1 (1950), i+24. MR 46641