finite topological $\mathcal L $-structures.

An *epimorphism* between topological $ \mathcal L $-structures $A, B$ is a continuous surjection $ {\varphi}:A\to B$ such that $$r^B=\underset{n \text{ times}}{\underbrace{ {\varphi}\times \ldots \times {\varphi}}}\, (r^A)$$ for every $n$-ary relation symbol $r \in {\mathcal{L}}$: in other words, $r^B(b_1,\ldots ,b_n)$ if and only if there exist $a_1,\ldots ,a_n\in A$ such that $${\varphi}(a_1)=b_1,\ldots , {\varphi}(a_n)=b_n,\quad r^A(a_1,\ldots ,a_n).$$ An *isomorphism* is a bijective epimorphism, so in particular it is a homeomorphism between the supports. An isomorphism of $A$ onto $A$ is an *automorphism* and we denote by $\operatorname{Aut}(A)$ the group of automorphisms of $A$. An epimorphism $ {\varphi}:A\to B$ *refines* a covering $ \mathcal U $ of $A$ if the preimage of any element of $B$ is included in some element of $ \mathcal U $. If $ \mathcal G , \mathcal G'
 $ are families of topological structures such that $ \mathcal G'
 \subseteq \mathcal G $ and for all $A\in \mathcal G$ there exist $ B\in \mathcal G'
$ and epimorphism ${\varphi}: B \to A $, we say that $ \mathcal G'
 $ is *cofinal* in $ \mathcal G $.

A family $ \mathcal G $ of topological $ \mathcal L $-structures is a *projective 