General position properties satisfied by finite products of dendrites
Author:
Philip L. Bowers
Journal:
Trans. Amer. Math. Soc. 288 (1985), 739753
MSC:
Primary 54F50; Secondary 54C25, 54C35, 54F35
MathSciNet review:
776401
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Abstract: Let be a dendrite whose endpoints are dense and let be the complement in of a dense compact collection of endpoints of . This paper investigates various general position properties that finite products of and possess. In particular, it is shown that (i) if is an space that satisfies the disjoint cells property, then satisfies the disjoint cells property, (ii) is a compact dimensional that satisfies the disjoint cells property, (iii) is a compact dimensional that satisfies the stronger general position property that maps of dimensional compacta into are approximable by both maps and embeddings, and (iv) is a topologically complete dimensional that satisfies the discrete cells property and as such, maps from topologically complete separable dimensional spaces into are strongly approximable by closed embeddings.
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 [Bo ]
 P. L. Bowers, Applications of general position properties of dendrites to Hilbert space topology, Ph.D. Dissertation, Univ. of Tennessee, 1983.
 [Bo ]
 , Discrete cells properties in the boundary set setting, Proc. Amer. Math. Soc. (to appear). MR 776212 (86d:57008)
 [Ca]
 J. W. Cannon, Shrinking celllike decompositions of manifolds. Codimension three, Ann. of Math. (2) 110 (1979), 83112. MR 541330 (80j:57013)
 [Ch]
 T. A. Chapman, Lectures on Hilbert cube manifolds, CBMS Regional Conf. Ser. in Math., no. 28, Amer. Math. Soc., Providence, R.I., 1976. MR 0423357 (54:11336)
 [Cu ]
 D. W. Curtis, Boundary sets in the Hilbert cube, preprint.
 [Cu ]
 , Preliminary report, boundary sets in the Hilbert cube and applications to hyperspaces, preprint.
 [Da]
 R. J. Daverman, Detecting the disjoint disks property, Pacific J. Math. 93 (1981), 277298. MR 623564 (82k:57007)
 [DW]
 R. J. Daverman and J. J. Walsh, Čech homology characterizations of infinite dimensional manifolds, Amer. J. Math. 103 (1981), 411435. MR 618319 (83k:57008)
 [DT]
 T. Dobrowolski and H. Torunczyk, On metric linear spaces homeomorphic to and compact convex subsets homeomorphic to , Bull. Acad. Polon. Sci. 27 (1979), 883886. MR 616181 (82j:57010)
 [Du]
 J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
 [Ed]
 R. D. Edwards, Approximating certain celllike maps by homeomorphisms, Abstract preprint. See also Notices Amer. Math. Soc. 24 (1977), A649, #751G5.
 [HW]
 W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N.J., 1969. MR 0006493 (3:312b)
 [vM]
 J. van Mill, A boundary set for the Hilbert cube containing no arcs, Fund. Math. (to appear). MR 732657 (86a:57012)
 [Qu]
 F. Quinn, Ends of maps and applications, Ann. of Math. (2) 110 (1979), 275331. MR 549490 (82k:57009)
 [Si]
 K. Sieklucki, A generalization of a theorem of K. Borsuk concerning the dimension of sets, Bull. Acad. Polon. Sci. 10 (1962), 433436. MR 0198430 (33:6588)
 [To ]
 H. Torunczyk, On images of the Hilbert cube and characterization of manifolds, Fund. Math. 106 (1980), 3140. MR 585543 (83g:57006)
 [To ]
 , Characterizing Hilbert space topology, Fund. Math. 111 (1981), 247262. MR 611763 (82i:57016)
 [Wi]
 S. Willard, General topology, AddisonWesley, Reading, Mass., 1970. MR 0264581 (41:9173)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507764015
PII:
S 00029947(1985)07764015
Keywords:
Disjoint cells property,
discrete cells property,
locally connected in
Article copyright:
© Copyright 1985
American Mathematical Society
