There are many different, and not necessarily equivalent, definitions of zero-dimensionality for a topological space. Here are two examples:
- Def. 1: A topological space is zero-dimensional if every open cover of the space has a refinement which is itself a cover consisting only of disjoint open sets. (This implies the Lebesgue covering dimension is zero.)
- Def. 2: A topological space is zero-dimensional if it has a base consisting exclusively of clopen sets. (This implies the small inductive dimension is zero.)
Suppose I were to create the following, perhaps more geometrically intuitive definition of a zero-dimensional topological space:
- Def. 3: A topological space is zero-dimensional if no subspace of the original space is homeomorphic to an interval equipped with the Euclidean topology.
What statements can we make about the connections, if any, between Def. 3 and Defs. 1 and 2? For example, does Def. 3 imply Def. 2 or 1? Does it imply it only in separable metrizable spaces? (And so on.)
Best Answer
There are separable metric complete spaces that are infinite-dimensional and are totally disconnected which implies definition 3. So that definition does not imply the other ones. 1 and 2 are equivalent for separable metric spaces of course ( and also for larger classes of spaces) and both imply 3. That’s about all one can say I think.