In this paper an example is
constructed of a compact Hausdorff space X with covering and large inductive
dimensions 0, but containing subsets Yn, 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 Yn are
discussed and it is shown that the covering dimension of βYn= 0 for 1 ≦ n < ∞. It
is known (see Gillman and Jerison) that for a normal space N, cov dimN = cov
dimβN.
