A predicate topological space is a pair \(\langle X,D\rangle\), where \(X\) is a topological space, \(D\) is a system of domains, i.e., a function assigning a non-empty domain \(D_x\) to each point \(x\in X\) in such a way that for any \(d\in D_X=\bigcup\limits_{x\in X}D_x\), the set \(\{x\in X:d\in D_x\}\) is open in \(X\). If \(D_x\) does not depend on \(x\), then \(\langle X,D\rangle\) is a predicate topological space with a constant domain. A predicate topological model based on a predicate topological space \(\langle X,D\rangle\) is a triple \(\mathbf M=\langle X,D,V\rangle\), where \(V\) is an evaluation, i.e., a function assigning an open set \(\mathrm{Val}_{\mathbf M}(A)\) to each atomic sentence \(A\) with constants in \(D_X\). Then \(\mathrm{Val}_{\mathbf M}(A)\) is defined for each sentence \(A\) according to the interpretation of logical operations and quantifiers in the Heyting algebra of the open subsets of \(X\). A pair \(\langle\Gamma,\Delta\rangle\) of non-empty sets of sentences is called consistent iff the formula \(\bigwedge\Gamma'\to\bigvee\Delta'\) is not a theorem of the intuitionistic predicate logic QH for any finite subsets \(\Gamma'\subseteq\Gamma\), \(\Delta'\subseteq\Delta\). One says that QH is strongly complete for a predicate topological space \(\langle X,D\rangle\) iff for any consistent pair \(\langle\Gamma,\Delta\rangle\), there exists a predicate topological model \(\mathbf M\) based on \(\langle X,D\rangle\) such that \(\bigcap_{A\in\Gamma}\mathrm{Val}_{\mathbf M}(A)\setminus\bigcup_{A\in\Delta} \mathrm{Val}_{\mathbf M}(A)\not=\emptyset\). QH is strongly complete for a topological space \(X\) iff there exists a system of domains \(D\) such that QH is strongly complete for \(\langle X,D\rangle\). It is known that QH is strongly complete for the irrational line \(\mathbb P\) and the rational line \(\mathbb Q\). Each of \(\mathbb P\) and \(\mathbb Q\) is a separable zero-dimensional dense-in-itself metrizable space. The main result of the paper is Theorem 3.1: QH is strongly complete for any zero-dimensional dense-in-itself metrizable space with a constant domain of cardinality \(\le\) the space’s weight, i.e., the minimal cardinality of a basis for the space. If varying domains are allowed, then the requirement of zero-dimensionality can be removed and only countable domains can be considered. Namely, Theorem 3.3 states that for any dense-in-itself metrizable space \(X\), there is a system of countable domains \(D\) such that QH is strongly complete for \(\langle X,D\rangle\). It follows that QH is strongly complete for \(\mathbb R\) in topological semantics with varying domains although this is not the case with a constant domain.