Abstract:
We prove that a dyadic space of weight $\tau$ contains the Cantor cube $D^\tau$ if and only if it cannot be represented as a countable union of closed subsets with weights less than $\tau$. A similar result has been independently obtained by Gerlits. That solves a problem posed by Pełczyǹski. In the particular case when the dyadic space is, in addition, a Dugundji space, the problem has been recently solved by Haydon. Further, it follows that any dyadic space whose weight $\tau$ is not a sum of countably many smaller cardinals can be continuously mapped onto the Tikhonov cube $I^\tau$ and contains the Cantor cube $D^\tau$. This is true, in particular, when $\tau$ is a regular cardinal, as was proved by Hagler. By means of the methods developed in this paper we prove that the depth of a dyadic space is equal to its cardinality and is attained; this is a final solution of Arkhangel'skii's problem about the “depth” of dyadic spaces.
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