On the thickness of topological spaces (Q1911505)

The author uses bases for a given topological space \(X\) and the usual definition for monad, \(\mu (a)\), defined on a local base \({\mathcal B}_a= \{B\in {\mathcal B}\mid a\in {}^*B\}\) about any point \(a\in {}^*X\) (i.e. ``the halo in base \({\mathcal B}\) of \(a\)'') to define a measure for the ``thickness'' of \(X\). This is done in steps, where the first step is to consider chains of such monads, \(\mu (a_p) \subset \dots \subset \mu(a_1) \subset \mu(^*x)\), \(x\in X\). This leads to the idea of the ``thickness in \(x\) of \({\mathcal B}\)'', the ``thickness of \({\mathcal B}\), \(\text{ep }{\mathcal B}\)'', and finally the ``thickness of \(X\), \(\text{ep } X\)''. The paper should be consulted for the complete definitions, but they are rather straightforward in character. The author then establishes certain characterizations. For example, if \(X\) is nonempty and \({\mathcal B}\) a base for \(X\), then \(\text{ep } {\mathcal B} =0\) if and only if \({\mathcal B}\) consists of open-closed subsets of \(X\). The major results are relative to how this definition compares to the small inductive dimension, ind the large inductive dimension, Ind, and the covering dimension, dim. The author shows that, in general, (a) \(\text{ep } X=0\) if and only if \(\text{ind } X=0\), and (b) \(\text{ind } X\leq \text{ep } X\). Along with other results relating \(\text{ep } X\) with these standard dimension concepts, the author shows that for every separable metric space \(X\), \(\text{ep } X= \text{ind } X= \text{Ind } X= \dim X\). On the other hand, there are metric spaces where \(\text{ep } X<\dim X\) and nonseparable metric spaces where \(\text{ep } X= \text{ind } X= \text{Ind } X= \dim X\).