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Local Isomorphisms of Algebras of Continuous Functions
Author(s) -
Goodearl K. R.
Publication year - 1977
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-16.2.348
Subject(s) - mathematics , homeomorphism (graph theory) , isomorphism (crystallography) , neighbourhood (mathematics) , invariant (physics) , pure mathematics , hausdorff space , combinatorics , mathematical analysis , crystal structure , crystallography , mathematical physics , chemistry
This paper is concerned with the problem of when a local isomorphism of C ( X ) and C ( Y ) implies a local homeomorphism of X and Y . More precisely, if X and Y are locally compact Hausdorff spaces, if p ɛ Y and q ɛ Y , and if C ( X ) P is isomorphic to C ( Y ) q , does p have a neighbourhood which is homeomorphic to a neighbourhood of q ? The answer is negative in general, but is shown to be positive in two cases: when X and Y are metric and the isomorphism of C ( X ) P and C ( Y ) q arises from a continuous map of Y into X , and when X and Y are manifolds. The proof for manifolds proceeds by showing that the local topological dimension of X at p is an invariant of the ring C ( X ) P , which is proved with the aid of the lattice of germs of closed sets at p .