Vol. 41, No. 3, 1972

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On the non-monotony of dimension

Beverly L. Brechner

Vol. 41 (1972), No. 3, 587–600

In this paper an example is constructed of a compact Hausdorff space X with covering and large inductive dimensions 0, but containing subsets Y n, with covering dimension n, and large inductive dimension at least n, each n,1 n . This result is an interesting contrast with the recent result of D. W. Henderson that there exists an infinite dimensional compact metric space with no n-dimensional compact subsets for 1 n < . As a corollary of our results, we show that for each n,1 n , there exists a (necessarily non-metric) continuum Mn of covering dimension n, which contains subsets of all positive covering dimensions. Covering dimension is treated in §2, while large inductive dimension is treated in §4. In §3, compactifications of Y n are discussed and it is shown that the covering dimension of βY n = 0 for 1 n < . It is known (see Gillman and Jerison) that for a normal space N, cov dimN = cov dimβN.

Mathematical Subject Classification 2000
Primary: 54F45
Received: 4 February 1971
Revised: 17 August 1971
Published: 1 June 1972
Beverly L. Brechner