On weakly infinite-dimensional subspuees

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Title

Authors ¹

Department of Mathematics, Free University, de Boelelaan 1081, 1081 Hv Amsterdam, The Netherlands

We will construct weakly infinite-dimensional (in the sense of Y. Smirnov) spaces X and Y such that Y contains X topologically and

dim Y = ω_{0}

and

dim X = ω_{0} + 1

. Consequently, the subspace theorem does not hold for the transfinite dimension dim for weakly infinite-dimensional spaces.

weakly infinite-dimensional, transfinite dimension

[B1] P. Borst, Classification of weakly infinite-dimensional spaces. Part I: A transfinite extension of the covering dimension, Fund. Math. 130 (1988), 1-25.
[B2] P. Borst, Classification of weakly infinite-dimensional spaces. Part II: Essential mappings, ibid., 73-99.
[Ch] V. A. Chatyrko, On the transfinite dimension dim, to appear.
[E1] R. Engelking, General Topology, PWN, Warszawa 1977.
[E2] R. Engelking, Dimension Theory, PWN, Warszawa 1978.
[He] D. W. Henderson, A lower bound for transfinite dimension, Fund. Math. 63 (1968), 167-173.
[L] L. A. Lyuksemburg, On transfinite inductive dimensions, Soviet Math. Dokl. 14 (1973), 388-393.
[P] R. Pol, On classification of weakly infinite-dimensional compacta, Fund. Math. 116 (1983), 169-188.

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