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1998 | 158 | 2 | 147-164
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Dugundji extenders and retracts on generalized ordered spaces

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For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an $L_{ch}$-extender (resp. $L_{cch}$-extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X\A; (iii) there is a continuous $L_{ch}$-extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and the pointwise convergence topology, for each space Y; (iv) A × Y is C*-embedded in X × Y for each space Y; (v) there is a continuous linear extender $φ:C*_{k}(A) → C_{p}(X)$; (vi) there is an $L_{ch}$-extender φ:C(A) → C(X); and (vii) there is an $L_{cch}$-extender φ:C(A) → C(X). We prove that these conditions are related as follows: (i)⇒(ii)⇔(iii)⇔(iv)⇔(v)⇒(vi)⇒(vii). If A is paracompact and the cellularity of A is nonmeasurable, then (ii)-(vii) are equivalent. If there is no connected subset of X which meets distinct convex components of A, then (ii) implies (i). We show that van Douwen's example of a separable GO-space satisfies none of the above conditions, which answers questions of Heath-Lutzer [9], van Douwen [1] and Hattori [8].
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Bibliografia
  • [1] E. K. van Douwen, Simultaneous extension of continuous functions, Ph.D. thesis, Free University of Amsterdam, 1975.
  • [2] E. K. van Douwen, Retracts of the Sorgenfrey line, Compositio Math. 38 (1979), 155-161.
  • [3] R. Engelking, On closed images of the space of irrationals, Proc. Amer. Math. Soc. 21 (1969), 583-586.
  • [4] R. Engelking, General Topology, revised and completed edition, Heldermann, Berlin, 1989.
  • [5] R. Engelking and D. Lutzer, Paracompactness in ordered spaces, Fund. Math. 94 (1977), 49-58.
  • [6] M. J. Faber, Metrizability in Generalized Ordered Spaces, Math. Centre Tracts 53, Math. Centrum, Amsterdam, 1974.
  • [7] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, 1960.
  • [8] Y. Hattori, π-embedding and Dugundji extension theorems for generalized ordered spaces, Topology Appl. 84 (1998), 43-54.
  • [9] R. W. Heath and D. J. Lutzer, Dugundji extension theorems for linearly ordered spaces, Pacific J. Math. 55 (1974), 419-425.
  • [10] R. W. Heath, D. J. Lutzer and P. L. Zenor, Monotonically normal spaces, Trans. Amer. Math. Soc. 178 (1973), 481-493.
  • [11] R. W. Heath, D. J. Lutzer and P. L. Zenor, On continuous extenders, in: Studies in Topology, N. M. Starakas and K. R. Allen (eds.), Academic Press, New York, 1975, 203-213.
  • [12] D. J. Lutzer, On generalized ordered spaces, Dissertationes Math. 89 (1977).
  • [13] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182.
  • [14] J. van Mill (ed.), Eric K. van Douwen Collected Papers, North-Holland, Amsterdam, 1994.
  • [15] K. Morita, On the dimension of the product of topological spaces, Tsukuba J. Math. 1 (1977), 1-6.
  • [16] T. C. Przymusiński, Product spaces, in: Surveys in General Topology, G. M. Reed (ed.), Academic Press, New York, 1980, 399-429.
  • [17] I. S. Stares and J. E. Vaughan, The Dugundji extension property can fail in $ω_μ$-metrizable spaces, Fund. Math. 150 (1996), 11-16.
  • [18] A. Waśko, Extensions of functions defined on product spaces, ibid. 124 (1984), 27-39.
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bwmeta1.element.bwnjournal-article-fmv158i2p147bwm
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