Perfect mappings and certain interior images of spaces
Authors:
J. M. Worrell and H. H. Wicke
Journal:
Trans. Amer. Math. Soc. 181 (1973), 2335
MSC:
Primary 54C10
MathSciNet review:
0320990
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Abstract: The main theorems of this paper show that certain conditions (called , and ) are invariant, in the presence of 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 open continuous images of some classes of 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 space is an open continuous image of a paracompact Čech complete space if and only if the space satisfies condition [Pacific J. Math. 37 (1971), 265275]. Moreover, in the same paper it is shown that if a completely regular space satisfies condition then any completely regular open continuous image of it also satisfies . 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 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 regularity is the subclass satisfying condition . Similar results are obtained for the regular spaces satisfying , and . The other classes of spaces involved are the regular complete spaces (i.e., spaces which are quasiperfect preimages of complete metric spaces), paracompact spaces, and regular spaces. In the last two cases besides the inferiority of the mappings the notion of uniform completeness, which generalizes compactness of a mapping, enters. (For details see General Topology and Appl. 1 (1971), 85100.) The proofs are accomplished through the use of two basic lemmas on closed continuous mappings satisfying certain additional conditions.
 [1]
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
(32 #8299)
 [2]
V.
V. Filippov, The perfect image of a paracompact feathery
space, Dokl. Akad. Nauk SSSR 176 (1967),
533–535 (Russian). MR 0222853
(36 #5903)
 [3]
M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 174.
 [4]
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
0125559 (23 #A2859)
 [5]
Paul
R. Halmos, Naive set theory, The University Series in
Undergraduate Mathematics, D. Van Nostrand Co., Princeton,
N.J.TorontoLondonNew York, 1960. MR 0114756
(22 #5575)
 [6]
Tadashi
Ishii, On closed mappings and 𝑀spaces. I, II, Proc.
Japan Acad. 43 (1967), 752–756; 757–761. MR 0222854
(36 #5904)
 [7]
John
L. Kelley, General topology, D. Van Nostrand Company, Inc.,
TorontoNew YorkLondon, 1955. MR 0070144
(16,1136c)
 [8]
D. König, Sur les correspondences multivoques, Fund. Math. 8(1926), 114134.
 [9]
E.
Michael, Another note on paracompact
spaces, Proc. Amer. Math. Soc. 8 (1957), 822–828. MR 0087079
(19,299c), http://dx.doi.org/10.1090/S00029939195700870799
 [10]
E.
Michael, A note on closed maps and compact sets, Israel J.
Math. 2 (1964), 173–176. MR 0177396
(31 #1659)
 [11]
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
(27 #709)
 [12]
Kiiti
Morita, Products of normal spaces with metric spaces, Math.
Ann. 154 (1964), 365–382. MR 0165491
(29 #2773)
 [13]
Kiiti
Morita, Some properties of 𝑀spaces, Proc. Japan Acad.
43 (1967), 869–872. MR 0227933
(37 #3517)
 [14]
Howard
H. Wicke, The regular open continuous images of complete metric
spaces, Pacific J. Math. 23 (1967), 621–625. MR 0219035
(36 #2118)
 [15]
Howard
H. Wicke, Open continuous images of certain kinds of
𝑀spaces and completeness of mappings and spaces, General
Topology and Appl. 1 (1971), no. 1, 85–100. MR 0282348
(43 #8060)
 [16]
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 0309078
(46 #8189)
 [17]
, On a class of spaces containing Arhangel' skiĭ's spaces, Notices Amer. Math. Soc. 14 (1967), 687. Abstract #648188.
 [18]
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 0307173
(46 #6294)
 [19]
J.
M. Worrell Jr. and H.
H. Wicke, Characterizations of developable topological spaces,
Canad. J. Math. 17 (1965), 820–830. MR 0182945
(32 #427)
 [20]
John
M. Worrell Jr., Upper semicontinuous decompositions of
developable spaces, Proc. Amer. Math. Soc.
16 (1965),
485–490. MR 0181982
(31 #6207), http://dx.doi.org/10.1090/S00029939196501819821
 [21]
J.
M. Worrell Jr., Upper semicontinuous decompositions of spaces
having bases of countable order, Portugal. Math. 26
(1967), 493–504. MR 0257977
(41 #2626)
 [22]
, A perfect mapping not preserving the space property, presented at Pittsburgh Conference on General Topology 1970.
 [1]
 A. V. Arhangel' skiĭ, On a class of spaces containing all metric and all locally compact spaces, Mat. Sb. 67 (109) (1965), 5588; English transl., Amer. Math. Soc. Transl. (2) 92 (1970), 139. MR 32 #8299; MR 42 #3. MR 0190889 (32:8299)
 [2]
 V. V. Filippov, On the perfect image of a paracompact space, Dokl. Akad. Nauk SSSR 176 (1967), 533535 = Soviet Math. Dokl. 8 (1967), 11511153. MR 36 #5903. MR 0222853 (36:5903)
 [3]
 M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 174.
 [4]
 Z. Frolík, On the topological product of paracompact spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 747750. MR 23 #A2859. MR 0125559 (23:A2859)
 [5]
 P. R. Halmos, Naive set theory, University Ser. in Undergraduate Math., VanNostrand, Princeton, N. J., 1960. MR 22 #5575. MR 0114756 (22:5575)
 [6]
 T. Ishii, On closed mappings and spaces. II., Proc. Japan Acad. 43 (1967), 757761. MR 36 #5904. MR 0222854 (36:5904)
 [7]
 J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
 [8]
 D. König, Sur les correspondences multivoques, Fund. Math. 8(1926), 114134.
 [9]
 E. Michael, Another note on paracompact spaces, Proc. Amer. Math. Soc. 8 (1957), 822828. MR 19, 299. MR 0087079 (19:299c)
 [10]
 , A note on closed maps and compact sets, Israel J. Math. 2 (1964), 173176. MR 31 #1659. MR 0177396 (31:1659)
 [11]
 R. L. Moore, Foundations of point set theory, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 13, Amer. Math. Soc., Providence, R. I., 1962. MR 27 #709. MR 0150722 (27:709)
 [12]
 K. Morita, Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365382. MR 29 #2773. MR 0165491 (29:2773)
 [13]
 , Some properties of spaces, Proc. Japan Acad. 43 (1967), 869872. MR 37 #3517. MR 0227933 (37:3517)
 [14]
 H. H. Wicke, The regular open continuous images of complete metric spaces, Pacific J. Math. 23 (1967), 621625. MR 36 #2118. MR 0219035 (36:2118)
 [15]
 , Open continuous images of certain kinds of spaces and completeness of mappings and spaces, General Topology and Appl. 1 (1971), 85100. MR 43 #8060. MR 0282348 (43:8060)
 [16]
 H. H. Wicke and J. M. Worrell, Jr., On topological completeness of first countable Hausdorff spaces. I, Fund. Math. 75 (1972), 209222. MR 0309078 (46:8189)
 [17]
 , On a class of spaces containing Arhangel' skiĭ's spaces, Notices Amer. Math. Soc. 14 (1967), 687. Abstract #648188.
 [18]
 , On the open continuous images of paracompact Čech complete spaces, Pacific J. Math. 37 (1971), 265275. MR 0307173 (46:6294)
 [19]
 J. M. Worrell, Jr. and H. H. Wicke, Characterizations of developable topological spaces, Canad. J. Math. 17 (1965), 820830. MR 32 #427. MR 0182945 (32:427)
 [20]
 J. M. Worrell, Jr., Upper semicontinuous decompositions of developable spaces, Proc. Amer. Math. Soc. 16 (1965), 485490. MR 31 #6207. MR 0181982 (31:6207)
 [21]
 , Upper semicontinuous decompositions of spaces having bases of countable order, Portugal. Math. 26 (1967), 493504. MR 41 #2626. MR 0257977 (41:2626)
 [22]
 , A perfect mapping not preserving the space property, presented at Pittsburgh Conference on General Topology 1970.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197303209903
PII:
S 00029947(1973)03209903
Keywords:
Čech complete spaces,
space,
space,
space,
space,
space,
complete space,
space,
pointcountable covering,
first countablelike closed continuous mappings,
primitive sequence,
monotonically contracting sequence,
(absolute) set of interior condensation,
perfect mappings,
open continuous mappings
Article copyright:
© Copyright 1973
American Mathematical Society
