There is no upper bound of small transfinite compactness degree in metrizable spaces |
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Authors: | Vitalij A Chatyrko |
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Institution: | a Department of Mathematics, Linkeping University, 581 83 Linkeping, Sweden b Department of Mathematics, Shimane University, Matsue, Shimane, 690-8504 Japan |
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Abstract: | In J.M. Aarts, T. Nishiura, Dimension and Extensions, North-Holland, Amsterdam, 1993], Aarts and Nishiura investigated several types of dimensions modulo a class P of spaces. These dimension functions have natural transfinite extensions. The small transfinite compactness degree trcmp is such transfinite dimension function extending the small compactness degree cmp. We shall prove that there is no upper bound for trcmp in the class of metrizable spaces, i.e. for each ordinal number α there exists a metrizable space Xα such that trcmpXα=α. We also give a characterization of the dimension dim of a separable (compact) metrizable space in terms of the function cmp of the product of this space with a nowhere locally compact zero-dimensional factor. |
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Keywords: | primary 54F45 secondary 54E35 |
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