which does not contain any nonempty clopen subset of $X$.

-   The *quasi-component* of a point is the intersection of all its clopen neighborhoods. A space is *totally separated* if the quasi-component of each point is a singleton.

-   A space is $n$-*homogeneous* if for every two sets of $n$ points there is a homeomorphism sending one onto the other.

-   A space $X$ is $\nicefrac{1}{n}$-*homogeneous* if the action of $\operatorname{Homeo}(X)$ on $X$ has exactly $n$ orbits.

-   A space is $h$-*homogeneous* if it is homeomorphic to each of its nonempty clopen subsets.

When we talk about dimension, we mean the inductive dimension.

Topological structures
----------------------

We recall here some basic definitions, mainly from [@Irwin2006; @Camerl2010], sticking to relational first order languages, since we will not use other kinds of languages in this paper.

Let thus a relational first order language $ \mathcal L $ be given. A *topological* $ \mathcal L $-*structure* is a zero-dimensional compact metrizable space that is also an $ \mathcal L $-structure such that the interpretations of the relation symbols are closed sets. In particular, the topology on finite topological $\mathcal L $-structures is discrete. We will usually suppress the word “topological” when referring to 