Environments of closed sets, and complements in metrizable spaces

AbstractFor nowhere dense closed subspaces S of X and T of Y, when does S sit in X the same way T sits in Y? We introduce environments to give a possible answer: when S and T have isomorphic environments. (S has an environment in X if X is metrizable, but not in general.)For metrizable S and T we show that having isomorphic environments in suitable enlargements is equivalent to the existence of a (metrizable) space which admits perfect maps onto X and Y