THE OPEN-OPEN TOPOLOGY FOR FUNCTION SPACES

KATHRYN F. PORTER

《International Journal of Mathematics and Mathematical Sciences》 1993卷 (1993/12) Pp. 111-116

https://doi.org/10.1155/S0161171293000134

摘要

Let (X, T) and (Y, T*) be topological spaces and let F ⊂ Y^X. For each U ∈ T, V ∈ T*, let (U, V)={f ∈ F: f(U)⊂V}. Define the set S_(oo)={(U, V):U ∈ T and V ∈ T*}. Then S_(oo) is a subbasis for a topology, T_(oo) on F, which is called the open-open topology. We compare T_(oo) with other topologies and discuss its properties. We also show that T_(oo), on H(X), the collection of all self-homeomorphisms on X, is equivalent to the topology induced on H(X) by the Pervin quasi-uniformity on X.

關鍵字

Compact-open topology ； admissible topology ； Galois space ； Pervin quasi-uniformity ； self-homeomorphism ； quasi-uniform convergence