On topological definitions of zero-dimensionality

Question

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.)

Answer 1

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.

Answer 2

1 and 2 easily imply 3, but the implication does not reverse, not even for compact metric spaces, since the pseudoarc satisfies 3 but is one dimensional.

1 and 2 are equivalent for separable metric spaces, since $\dim X=\operatorname{ind}X=\operatorname{Ind} X$ for separable metric $X$.

In general consider the following three statements:

a. $\dim X=0$,
b. $X$ has a clopen basis,
c. $X$ is totally disconnected.

For $T_1$ spaces we have $a\implies b\implies c$ and for compact Hausdorff spaces all three are equivalent. In your numbering we thus have 1 implies 2 for all $T_1$ spaces.