Perfect mappings and certain interior images of $M$-spaces
HTML articles powered by AMS MathViewer
- by J. M. Worrell and H. H. Wicke PDF
- Trans. Amer. Math. Soc. 181 (1973), 23-35 Request permission
Abstract:
The main theorems of this paper show that certain conditions (called ${\lambda _c},{\lambda _b},{\beta _c}$, and ${\beta _b}$) are invariant, in the presence of ${T_0}$-regularity, under the application of closed continuous peripherally compact mappings. Interest in these conditions lies in the fact that they may be used to characterize certain regular ${T_0}$ open continuous images of some classes of $M$-spaces in the sense of K. Morita, and in the fact that they are preserved by open continuous mappings with certain appropriate additional conditions. For example, the authors have shown that a regular ${T_0}$-space is an open continuous image of a paracompact Čech complete space if and only if the space satisfies condition ${\lambda _b}$ [Pacific J. Math. 37 (1971), 265-275]. Moreover, in the same paper it is shown that if a completely regular ${T_0}$-space satisfies condition ${\lambda _b}$ then any ${T_0}$ completely regular open continuous image of it also satisfies ${\lambda _b}$. These results together with the results of the present paper and certain known results lead to the following theorem: The smallest subclass of the class of regular ${T_0}$-spaces which contains all paracompact Čech complete spaces and which is closed with respect both to the application of perfect mappings and to the application of open continuous mappings preserving ${T_0}$-regularity is the subclass satisfying condition ${\lambda _b}$. Similar results are obtained for the regular ${T_0}$-spaces satisfying ${\lambda _c},{\beta _b}$, and ${\beta _c}$. The other classes of $M$-spaces involved are the regular ${T_0}$ complete $M$-spaces (i.e., spaces which are quasi-perfect preimages of complete metric spaces), ${T_2}$ paracompact $M$-spaces, and regular ${T_0}M$-spaces. In the last two cases besides the inferiority of the mappings the notion of uniform $\lambda$-completeness, which generalizes compactness of a mapping, enters. (For details see General Topology and Appl. 1 (1971), 85-100.) The proofs are accomplished through the use of two basic lemmas on closed continuous mappings satisfying certain additional conditions.References
- A. V. Arhangel′skiĭ, A class of spaces which contains all metric and all locally compact spaces, Mat. Sb. (N.S.) 67 (109) (1965), 55–88 (Russian). MR 0190889
- V. V. Filippov, The perfect image of a paracompact feathery space, Dokl. Akad. Nauk SSSR 176 (1967), 533–535 (Russian). MR 0222853 M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1-74.
- Z. Frolík, On the topoligical product of paracompact spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 747–750 (English, with Russian summary). MR 125559
- Paul R. Halmos, Naive set theory, The University Series in Undergraduate Mathematics, D. Van Nostrand Co., Princeton, N.J.-Toronto-London-New York, 1960. MR 0114756
- Tadashi Ishii, On closed mappings and $M$-spaces. I, II, Proc. Japan Acad. 43 (1967), 752–756; 757–761. MR 222854
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144 D. König, Sur les correspondences multivoques, Fund. Math. 8(1926), 114-134.
- E. Michael, Another note on paracompact spaces, Proc. Amer. Math. Soc. 8 (1957), 822–828. MR 87079, DOI 10.1090/S0002-9939-1957-0087079-9
- E. Michael, A note on closed maps and compact sets, Israel J. Math. 2 (1964), 173–176. MR 177396, DOI 10.1007/BF02759940
- 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
- Kiiti Morita, Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365–382. MR 165491, DOI 10.1007/BF01362570
- Kiiti Morita, Some properties of $M$-spaces, Proc. Japan Acad. 43 (1967), 869–872. MR 227933
- Howard H. Wicke, The regular open continuous images of complete metric spaces, Pacific J. Math. 23 (1967), 621–625. MR 219035, DOI 10.2140/pjm.1967.23.621
- Howard H. Wicke, Open continuous images of certain kinds of $M$-spaces and completeness of mappings and spaces, General Topology and Appl. 1 (1971), no. 1, 85–100. MR 282348, DOI 10.1016/0016-660X(71)90114-0
- H. H. Wicke and J. M. Worrell Jr., Topological completeness of first countable Hausdorff spaces. I, Fund. Math. 75 (1972), no. 3, 209–222. MR 309078, DOI 10.4064/fm-75-3-209-222 —, On a class of spaces containing Arhangel’ skiĭ’s $p$-spaces, Notices Amer. Math. Soc. 14 (1967), 687. Abstract #648-188.
- H. H. Wicke and J. M. Worrell Jr., On the open continuous images of paracompact Čech complete spaces, Pacific J. Math. 37 (1971), 265–275. MR 307173, DOI 10.2140/pjm.1971.37.265
- J. M. Worrell Jr. and H. H. Wicke, Characterizations of developable topological spaces, Canadian J. Math. 17 (1965), 820–830. MR 182945, DOI 10.4153/CJM-1965-080-3
- John M. Worrell Jr., Upper semicontinuous decompositions of developable spaces, Proc. Amer. Math. Soc. 16 (1965), 485–490. MR 181982, DOI 10.1090/S0002-9939-1965-0181982-1
- J. M. Worrell Jr., Upper semicontinuous decompositions of spaces having bases of countable order, Portugal. Math. 26 (1967), 493–504. MR 257977 —, A perfect mapping not preserving the $p$-space property, presented at Pittsburgh Conference on General Topology 1970.
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 181 (1973), 23-35
- MSC: Primary 54C10
- DOI: https://doi.org/10.1090/S0002-9947-1973-0320990-3
- MathSciNet review: 0320990