Homeomorphisms

Definition

Homeomorphisms are the isomorphisms in the category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms ${\textstyle X\to X}$ forms a group, called the homeomorphism group of X, often denoted ${\textstyle \operatorname {Homeo} (X).}$ This group can be given a topology, such as the compact-open topology, which under certain assumptions makes it a topological group.