Theorems on the Representability of Spaces as Unions of at Most Countably Many Homogeneous Subspaces

A topological space \(X\) is said to be homogeneous if for any \(x, y\in X\) there exists a self-homeomorphism \(f\) of \(X\) such that \(f(x)=y\) . We propose a method for constructing topological spaces representable as a union of \(n\) but not fewer homogeneous subspaces, where \(n\) is an arbitrary given positive integer. Further, we present a solution of a similar problem for the case of infinitely many summands.