Geodesic
Embedding inverse semigroups of homeomorphisms on closed subsets
Glasgow mathematical journal, Tome 18 (1977) no. 2, pp. 199-207

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

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X. 
Baird, Bridget Bos. Embedding inverse semigroups of homeomorphisms on closed subsets. Glasgow mathematical journal, Tome 18 (1977) no. 2, pp. 199-207. doi: 10.1017/S0017089500003281
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