We consider a natural way of extending the Lebesgue covering dimension to various classes of infinite dimensional separable metric spaces. All spaces in this paper are assumed to be separable metric spaces. Infinite games can be used in a natural way to define ordinal valued functions on the class of separable metric spaces. One of our examples of such an ordinal function coincides in any finit...

The Hausdorff dimension of a product X × Y can be strictly greater than that of Y , even when the Hausdorff dimension of X is zero. But when X is countable, the Hausdorff dimensions of Y and X × Y are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than “being countable” and stronger than “having Hausdorff dimension zero”. Fremlin asked whether it ...

We study dimension theory for the C∗-algebras of row-finite k-graphs with no sources. We establish that strong aperiodicity—the higher-rank analogue of condition (K)—for a k-graph is necessary and sufficient for the associated C∗-algebra to have topological dimension zero. We prove that a purely infinite 2-graph algebra has real-rank zero if and only if it has topological dimension zero and sat...

Let g be a polynomial automorphism of C. We study the dimensions (Hausdorff dimension and topological dimension) of the Julia set of g. We show that when g is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly greater than its topological dimension. Moreover, the Julia set cannot be locally connected. We also provide estimates for the dimension of the Julia sets in ...

The standard topological approach to indistinguishable particles formulates exchange statistics by using the fundamental group analyze connectedness of configuration space. Although successful in two and more dimensions, this gives only trivial or near one dimension because two-body coincidences are excluded from Instead, we include these path-ambiguous singular points consider space as an orbi...

The notion of compatible apparition points is introduced for non-Hausdorff manifolds, and properties of these points are studied. It is well known that the Hausdorff property is independent of the other conditions given in the standard definition of a topological manifold. In much of literature, a topological manifold of dimension n is a Hausdorff topological space which has a countable base of...