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The compact neighborhood extension property and local equi-connectedness

Authors: Nguyen To Nhu and Katsuro Sakai
Journal: Proc. Amer. Math. Soc. 121 (1994), 259-265
MSC: Primary 54C20; Secondary 54C55, 54D45, 54H11
MathSciNet review: 1232141
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Abstract: It is shown that any $ \sigma $-compact metrizable space is an AR (ANR) if and only if it is (locally) equi-connected and has the compact (neighborhood) extension property.

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  • [AE] R. Arens and J. Eells, On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397-403. MR 0081458 (18:406e)
  • [BM] J. van der Bijl and J. van Mill, Linear spaces, absolute retracts and the compact extension property, Proc. Amer. Math. Soc. 104 (1988), 942-952. MR 964878 (89k:57037)
  • [CM] D. W. Curtis and J. van Mill, The compact extension property, Proc. 6-th Prague Topology Sympos. (Z. Frolík, ed.), Hildermann Verlag, Berlin, 1988, pp. 115-119. MR 952598 (89k:54043)
  • [Do] T. Dobrowolski, On extending mapping into nonlocally convex linear metric spaces, Proc. Amer. Math. Soc. 93 (1985), 555-560. MR 774022 (86e:54023)
  • [Dr] A. N. Dranishnikov, On a problem of P. S. Alexandorff, Mat. Sb. 135 (1988), 551-557; English transl. Math. USSR-Sb. 63 (1989), 539-545. MR 942139 (90e:55004)
  • [E] R. D. Edwards, A theorem and a question related to cohomological dimension and cell-like mappings, Notices Amer. Math. Soc. 25 (1978), A259-A260.
  • [G] J. H. Gresham, A class of infinite-dimensional space. Part II: An Extension Theorem and the theory of retracts, Fund. Math. 106 (1980), 237-245. MR 585553 (81k:54063)
  • [Ha] W. E. Haver, Locally contractible spaces that are absolute neighborhood retracts, Proc. Amer. Math. Soc. 40 (1973), 280-284. MR 0331311 (48:9645)
  • [Hu] S.-T. Hu, Theory of retracts, Wayne St. Univ. Press, Detroit, 1965. MR 0181977 (31:6202)
  • [K] K. Kuratowski, Sur quelques problèmes topologique concernant le prolongement des fonctions continus, Colloq. Math. 2 (1951), 186-191. MR 0048791 (14:70d)
  • [vM] J. van Mill, Another counter-example in ANR-theory, Proc. Amer. Math. Soc. 97 (1986), 136-138. MR 831402 (87e:55001)
  • [N] Nguyen To Nhu, Investigating the ANR-property of metric spaces, Fund. Math. 124 (1984), 243-254; Corrections ibid. 141 (1992), 297. MR 774515 (86d:54018)
  • [NT] Nguyen To Nhu and Ta Khac Cu, On probability measure functions preserving the ANR-property of metric spaces, Proc. Amer. Math. Soc. 106 (1989), 493-501. MR 964459 (89m:60013)
  • [T] H. Toruńczyk, A short proof of Hausdorff's theorem on extending metrics, Fund. Math. 77 (1972), 191-193. MR 0321026 (47:9559)
  • [Wa] J. Walsh, Dimension, cohomological dimension, and cell-like mappings, Shape Theory and Geometric Topology--Proc., Dubrovnik 1981 (S. Mardešić and J. Segal, eds.), Lecture Notes in Math., vol. 870, Springer-Verlag, Berlin, 1981, pp. 105-118. MR 643526 (83a:57021)
  • [We] J. E. West, Open problems in infinite-dimensional topology, Open Problems in Topology (J. van Mill and G. M. Reed, eds.), Elsevier Sci. Publ. B.V. (North-Holland), Amsterdam, 1990, pp. 523-597. MR 1078666

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Keywords: The compact (neighborhood) extension property, AR, ANR, (locally) equi-connected
Article copyright: © Copyright 1994 American Mathematical Society

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