File:Topological space examples.svg
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Summary
Examples and non-examples of topological spaces, based roughly on Figures 12.1 and 12.2 from Munkres' Introduction to Topology. The 6 examples are subsets of the power set of {1,2,3}, with the small circle in the upper left of each denoting the empty set, and in reading order they are:
- {{}, {1,2,3}} (the trivial topology)
- {{}, {1}, {1,2,3}}
- {{}, {1}, {2}, {1,2}, {1,2,3}}
- {{}, {2}, {1,2}, {2,3}, {1,2,3}}
- {{}, {2}, {3}, {1,2,3}} (not a topology, because {2} and {3} are in the topology but their union {2,3} is not)
- {{}, {1,2}, {2,3}, {1,2,3}} (not a topology, because {1,2} and {2,3} are in the topology but their intersection {2} is not)
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 15:24, 12 January 2017 | | 360 × 319 (12 KB) | 127.0.0.1 (talk) | Examples and non-examples of topological spaces, based roughly on Figures 12.1 and 12.2 from Munkres' <i>Introduction to Topology</i>. The 6 examples are subsets of the power set of {1,2,3}, with the small circle in the upper left of each denoting the empty set, and in reading order they are: <ul> <li> {{}, {1,2,3}} (the trivial topology)</li> <li> {{}, {1}, {1,2,3}}</li> <li> {{}, {1}, {2}, {1,2}, {1,2,3}}</li> <li> {{}, {2}, {1,2}, {2,3}, {1,2,3}}</li> <li> {{}, {2}, {3}, {1,2,3}} (not a topology, because {2} and {3} are in the topology but their union {2,3} is not)</li> <li> {{}, {1,2}, {2,3}, {1,2,3}} (not a topology, because {1,2} and {2,3} are in the topology but their intersection {2} is not)</li> </ul> |
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