The open-open topology for function spaces

Kathryn F. Porter

International Journal of Mathematics and Mathematical Sciences, 1993, vol. 16, 1-6

Abstract:

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 ∘ ∘ = { ( U , V ) : U ∈ T  and  V ∈ T * } . Then S ∘ ∘ is a subbasis for a topology, T ∘ ∘ on F , which is called the open-open topology. We compare T ∘ ∘ with other topologies and discuss its properties. We also show that T ∘ ∘ , 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 .

Date: 1993
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:642372

DOI: 10.1155/S0161171293000134

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