Continuity and separation in symmetric topologies. (English) Zbl 1264.97009
Summary: In this note, it is shown that in a symmetric topological space, the pairs of sets separated by the topology determine the topology itself. It is then shown that when the codomain is symmetric, functions which separate only those pairs of sets that are already separated are continuous, generalizing a result found by M. Lynch [“Characterizing continuous functions in terms of separated sets”, ibid. 36, No. 5, 549–551 (2005)].
MSC:
| 97I99 | Analysis education |
| 54A05 | Topological spaces and generalizations (closure spaces, etc.) |
| 54C05 | Continuous maps |
| 54D65 | Separability of topological spaces |
References:
| [1] | DOI: 10.1080/00207390412331336210 · doi:10.1080/00207390412331336210 |
| [2] | Murdeshwar MG, General Topology (1983) |
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