Compactifications whose remainders are zero dimensional and retract (Q1373427)

In the following \(X\) denotes a locally compact noncompact Hausdorff space and \(fX\) is the Freudenthal compactification of \(X\). The first objective of this paper to provide a characterization of when \(fX\smallsetminus X\) is a retract of \(fX\). To this end, \(F(X)\) is defined to be the algebra of continuous real valued functions \(g\) on \(X\) for which there exists an open subset \(V\) of \(X\) having compact closure in \(X\) such that \(f(X\smallsetminus V)\) is finite. It is shown that \(fX\smallsetminus X\) is a retract of \(fX\) if and only if there exists a linear ring homomorphism \(P:F(X)\to F(X)\) such that \(P^2=P\), the image of the constant function 1 is itself and \((I-P) (F(X))= I(X)\), where \(I(X)\) is the ideal of elements \(g\) of \(F(X)\) for which \(X\smallsetminus g^{-1}(0)\) has compact closure in \(X\) and \(I:F(X)\to F(X)\) is the identity functions. Let 2 be the discrete two-point space, \(m\) the cardinal number of a set \(J\) and \(2^J\) the product of \(m\) copies of 2. The second theorem of this paper shows that for \(X\) locally compact and zero-dimensional, \(X\) as a dyadic family of power \(m\) if and only if there exists a compactification \(Y\) of \(X\) such that \(Y\smallsetminus X=2^J\) and \(Y\smallsetminus X\) is a retract of \(Y\).