On Menger algebras (Q1082637)

A pair (A,\(\phi\) : \(A\times A^ n\to A)\) is called a Menger algebra if \(\phi (\phi (x,Y),Z)=\phi (x,\Phi (Y,Z))\), where \(\Phi =\phi^ n: A^ n\times A^ n\to A^ n\) is the canonical extension of \(\phi\). If X is a topological space, \(A=C(X^ n,X)\) is the seet of continuous maps, \(A^ n\) is identified with \(C(X^ n)\), and \(\phi\) is composition, then \(M(X)=(A,\phi)\) is a Menger algebra. A \(T_ 1\)-space X is admissible, if \(\{f^{\leftarrow}(x): x\in X\), \(f\in C(X)\}\) is a basis for the closed sets. Theorem: Assume that X and Y are admissible, X is doubly homogeneous, and \(| Y| >1\). If there exists an epimorphism \(\eta\) : M(X)\(\to M(Y)\), then X and Y are homeomorphic.