Are there zero-dimensional topological spaces and are they not discrete?

Question

Let $\mathcal{T}$ be a topological space. Are there zero-dimensional subsets of $\mathcal{T}$ and are they not discrete?

My question is about Noetherian spaces, but I would like to know in the general case too.

Answer 1

In general topology zero-dimensional spaces are quite common:

Discrete spaces are zero-dimensional and products of zero-dimensional spaces are again zero-dimensional, so spaces like $\{0,1\}^\omega$ (which is homeomorphic to the Cantor set, $\{0,1\}$ is discrete) and $\mathbb{N}^\mathbb{N}$ (homeomorphic to the irrationals in $\mathbb{R}$, $\mathbb{N}$ has the discrete topology, like it has as a real subspace) are zero-dimensional subspaces, as well as all subspaces of zero-dimensional spaces. $\mathbb{Q}$ is also a non-discrete zero-dimensional space, as are $\omega+1 \simeq \{0\} \cup\{\frac1n: n \in \mathbb{N}^+\}\subseteq \mathbb{R}$, as well as all ordinals in the order topology. The Sorgenfrey line is also a famous example. A whole class of spaces is given by the Stone spaces of Boolean algebras: every Boolean algebra $A$ has an associated zero-dimensional space $X$, such that $A$ is isomorphic to the set of clopen sets of $X$. So in general topology, zero-dimensional spaces are widely studied and form a large class.

In fact, one can show that $X$ is zero-dimensional and Hausdorff iff there is some set $I$ such that $X$ is homeomorphic to a subspace of the so-called Cantor cube $\{0,1\}^I$.

I cannot offhand think of any examples of zero-dimensional Noetherian spaces, besides the finite discrete spaces. There might not be any other examples than that? At least $T_1$ ones: if we don't care about separation axioms we can take an indiscrete space or more generally any space such that the topology is a finite partition of the underlying set. (finite to get Noetherian, partition to get zero-dimensional, as all opens are then closed too.)