The Roelcke compactification of groups of homeomorphisms

✍ Scribed by V.V. Uspenskij

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 108 KB
• Volume: 111
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00185-6

✦ Synopsis

Let X be a zero-dimensional compact space such that all non-empty clopen subsets of X are homeomorphic to each other, and let Aut X be the group of all self-homeomorphisms of X, equipped with the compact-open topology. We prove that the Roelcke compactification of Aut X can be identified with the semigroup of all closed relations on X whose domain and range are equal to X. We use this to prove that the group Aut X is topologically simple and minimal.