Determining topologies by collections of proper maps and by weak topologies
| Autor: | C.L. Cooper |
|---|---|
| Jazyk: | angličtina |
| Předmět: |
Discrete mathematics
Weak topology C∞ manifold Extension topology Euclidean topology Initial topology Lower limit topology Compactly generated space C1 manifold Combinatorics Fixing a topology Product topology Geometry and Topology General topology Particular point topology First countable topology Mathematics |
| Zdroj: | Topology and its Applications. (3):203-213 |
| ISSN: | 0166-8641 |
| DOI: | 10.1016/0166-8641(94)90037-X |
| Popis: | A collection F of proper maps into a locally compact Hausdorff space (X,τ) is said to fix the topology τ if the only locally compact Hausdorff topology for X for which every map in the collection is continuous and proper is τ. It will be shown that the collection of C∞ injective paths fix the manifold topology of a compact manifold and the collection of analytic injective paths does not fix the manifold topology of any manifold. A related but different notion is that of determining a topology by means of a weak topology generated by a collection of subspaces. Specially, if C is a collection of subspaces of a topological space (X,τ) then C determines the topology on X if and only if the weak topology generated by C is the same as the topology τ. It will be shown that if (X,τ) is a first countable topology and if C is a collection of closed subspaces for which given any convergent sequence in (X,τ) there is a set in the collection C which contains a subsequence of the given sequence, then the collection determines the topology. |
| Databáze: | OpenAIRE |
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