Geodesic
Inverse Semigroups of Homeomorphisms are Hopfian
Canadian journal of mathematics, Tome 31 (1979) no. 4, pp. 800-807

Voir la notice de l'article provenant de la source Cambridge University Press

If X is a nonempty topological T1 space then the set of all homeomorphisms whose domains and ranges are closed subsets of X forms a semigroup under partial composition of functions. We call it IF(X). If, in a semigroup, every element a is matched with a unique element b such that aba = a and bab = b then the semigroup is an inverse semigroup (b is called the inverse of a and is denoted by a −1). We have that IF(X) is an inverse semigroup with the algebraic inverse of a map ƒ being just the inverse map ƒ -1. In this paper we examine epimorphisms from IF(X) onto IF(Y). The main theorem gives conditions under which an epimorphism must be an isomorphism. 
Baird, Bridget B. Inverse Semigroups of Homeomorphisms are Hopfian. Canadian journal of mathematics, Tome 31 (1979) no. 4, pp. 800-807. doi: 10.4153/CJM-1979-073-5
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