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Inverse semigroups of homeomorphisms between open subsets

Abstract The symbol IG(X) denotes the inverse semigroup, under composition of functions, of all homeomorphisms between open subsets of a T1 topological space X. The first result is that two such semigroups IG(X) and IG(Y) are isomorphic if and only if the spaces X and Y are homeomorphic. Ideals of IG(X) are next examined and it is shown that for many spaces X the semigroup IG(X) is 0-simple. We also look at congruences on IG(X); one result is that we determine a congruence which in many instances is the largest proper congruence on IG(X).