The open-open topology for function spaces
Let (X,T) and (Y,T*) be topological spaces and let F⊂YX. 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...
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| Format: | Article |
| Language: | English |
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Wiley
1993-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171293000134 |
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| _version_ | 1832551831312531456 |
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| author | Kathryn F. Porter |
| author_facet | Kathryn F. Porter |
| author_sort | Kathryn F. Porter |
| collection | DOAJ |
| description | Let (X,T) and (Y,T*) be topological spaces and let F⊂YX. 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. |
| format | Article |
| id | doaj-art-c3075c46484142fead0e9c6c368f0ece |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1993-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-c3075c46484142fead0e9c6c368f0ece2025-02-03T06:00:32ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251993-01-0116111111610.1155/S0161171293000134The open-open topology for function spacesKathryn F. Porter0Department of Mathematical Sciences, Saint Mary's College of California, Moraga, CA. 94575, USALet (X,T) and (Y,T*) be topological spaces and let F⊂YX. 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.http://dx.doi.org/10.1155/S0161171293000134compact-open topologyadmissible topologyGalois space Pervin quasi-uniformityself-homeomorphismquasi-uniform convergence. |
| spellingShingle | Kathryn F. Porter The open-open topology for function spaces International Journal of Mathematics and Mathematical Sciences compact-open topology admissible topology Galois space Pervin quasi-uniformity self-homeomorphism quasi-uniform convergence. |
| title | The open-open topology for function spaces |
| title_full | The open-open topology for function spaces |
| title_fullStr | The open-open topology for function spaces |
| title_full_unstemmed | The open-open topology for function spaces |
| title_short | The open-open topology for function spaces |
| title_sort | open open topology for function spaces |
| topic | compact-open topology admissible topology Galois space Pervin quasi-uniformity self-homeomorphism quasi-uniform convergence. |
| url | http://dx.doi.org/10.1155/S0161171293000134 |
| work_keys_str_mv | AT kathrynfporter theopenopentopologyforfunctionspaces AT kathrynfporter openopentopologyforfunctionspaces |