Counting ultrametric spaces
Volume 152 / 2018
Abstract
Due to the availability of connectedness arguments (using cut points) for proving spaces to be nonhomeomorphic, the enumeration problem for uncountable completely metrizable spaces has been solved under the restriction that only connected spaces are considered. Under this restriction, spaces of size smaller than $2^{\aleph _0}$ must fall out of consideration. This motivates us to solve the enumeration problem in the realm of totally disconnected spaces: For every infinite cardinal $\kappa $ we construct three families $\mathcal {F},\mathcal {G}_1,\mathcal {G}_2$ of mutually nonhomeomorphic strongly zero-dimensional completely metrizable spaces of weight $\kappa $ such that $|\mathcal {F}|=|\mathcal {G}_1|=|\mathcal {G}_2|=2^\kappa $ and all spaces in $\mathcal {F}$ are scattered (and hence of size $\kappa $), while all spaces in $\mathcal {G}_1\cup \mathcal {G}_2$ are perfect with $|X|=\max\{\kappa ,2^{\aleph _0}\}$ when $X\in \mathcal {G}_1$ and $|X|=\kappa ^{\aleph _0}$ when $X\in \mathcal {G}_2$. Furthermore, we show that for arbitrary infinite cardinals $\kappa ,\lambda $ with $\max\{\aleph _1,\kappa \}\leq \lambda \leq \kappa ^{\aleph _0} $ there exist $2^\lambda $ mutually nonhomeomorphic strongly zero-dimensional metrizable spaces of first category of weight $\kappa $ and size $\lambda $. As an application of this enumeration theorem, we also prove the existence of $2^\lambda $ mutually nonhomeomorphic connected, totally pathwise disconnected, nowhere locally connected metrizable spaces of weight $\kappa $ and size $\lambda $ without cut points whenever $ \max\{2^{\aleph _0},\kappa \}\leq \lambda \leq \kappa ^{\aleph _0}$.
