Local homeomorphisms of Stone's compact and local convertibility measures mappings

The paper is about open continuous mappings from extremely disconnected Hausdorf's compact of countable type into topological space with connectivity components that are not sets of the Baire first category. It is proved that such mapping is local homomorphism if and only if it maps all first-cathegory sets (maybe, except subsets of unique closed nowhere dense set) into first-cathegory sets. The obtained result is used for characterization of local reversibility of measurable mappings that act on standard spaces with measures. In particular, it is found out that Luzin’s $N$-condition does not only guarantee the measurability of an image but actually is also a criterion of local reversibility.