Environments of closed sets, and complements in metrizable spaces (Q689598)

This paper is motivated by the following question: If \(S\) is closed in \(X\) and \(T\) is closed in \(Y\), when does \(S\) sit in \(X\) the same way \(T\) sits in \(Y\)? In the special case when \(X\) and \(Y\) are \(Z\)-sets in the Hilbert cube, one answer is given by Chapman's complement theorem. In this paper, the author introduces the notion of environment and gives other possible answers in terms of isomorphisms of environments. For metrizable \(S\) and \(T\) he shows that having isomorphic environments in suitable enlargements is equivalent to the existence of a (metrizable) space which admits perfect maps onto \(X\) and \(Y\).