A topological space is zero dimensional if, and only if, it has a basis consisting of sets which are both open and closed (that is, "clopen").
This definition is according to "Counterexamples in Topology" by Steen and Seebach, 2nd ed. 1978.
I see that "zero dimensional" is tied in with various levels of being "disconnected", e.g. extremally disconnected, totally separated, totally, disconnected, scattered, etc. (although the property of being "zero dimensional" is itself strictly independent of all disconnectivity properties).
I understand (vaguely) the concept of a "dimension" in the context of manifolds: a space of $n$ dimensions has a boundary of $n – 1$ dimensions. I also note that a set is clopen if, and only if, it has no boundary. So would that be the basis of this terminology? Is there a source which states this definitively?
Best Answer
This notion of zero-dimensionality is based on the small inductive dimension $\operatorname{ind}(X)$ which can be defined for lots of spaces. It's defined inductively, based on the intuition that the boundary of open sets in two-dimensional spaces (which are boundaries disks, so circles in the plane, e.g.) have a dimension that is one lower than that of the space itself (this works nicely, at least intuitively, for the Euclidean spaces), setting up a recursion: We define $\operatorname{ind}(X) = -1$ iff $X=\emptyset$ (!) and a space has $\operatorname{ind}(X) \le n$ whenever $X$ has a base $\mathcal{B}$ of open sets such that $\operatorname{ind}(\partial O) \le n-1$ for all $O \in \mathcal{B}$, where $\partial A$ denotes the boundary of a set $A$. Finally, $\operatorname{ind}(X) = n$ holds if $\operatorname{ind}(X) \le n$ holds and $\operatorname{ind}(X)\le n-1$ does not hold. Also, $\operatorname{ind}(X)=\infty$ if for no $n$ we have $\operatorname{ind}(X) \le n$. It's clear that this is a topological invariant (homeomorphic spaces have the same dimension w.r.t. $\operatorname{ind}$) and zero-dimensional (i.e. $\operatorname{ind}(X)=0$) exactly means that there is a base of open sets with empty boundary (from the $-1$ clause!) and $\partial O=\emptyset$ means that $O$ is clopen.
Note that $\Bbb Q$ and $\Bbb P = \Bbb R\setminus \Bbb Q$ are both zero-dimensional in $\Bbb R$ and that $\Bbb R$ is one-dimensional (i.e. $\operatorname{ind}(\Bbb R)=1$) as boundaries of $(a,b)$ are $\{a,b\}$, which is zero-dimensional (discrete), etc.
In dimension theory more dimension functions have been defined as well, e.g. large inductive dimension $\operatorname{Ind}(X)$, which is a variant of $\operatorname{ind}(X)$, and the (Lebesgue) covering dimension $\dim(X)$, which has a different flavour and is about refinements of open covers and the order of those covers. For separable metric spaces however it can be shown that all $3$ that were mentioned are the same (i.e. give the same (integer) value). There are also metric-based definitions (fractal dimensions) which have more possible values, but are not topological, but metric invariants. Outside of metric spaces, we can have gaps between the dimension functions and stuff gets hairy quickly. See Engelking's book "Theory of dimensions, finite and infinite" for a taste of this field.
So in summary: the name "comes" (can be justified) from the small inductive dimension definition $\operatorname{ind}(X)$, but the name itself for that special class (clopen base) is older I think, and other names (like Boolean space etc) have been used too. It's a nice way to be very disconnected, giving a lot of structure.