indiscrete topology
If is a set and it is endowed with a topology defined by
then is said to have the indiscrete topology.
Furthermore is the coarsest topology a set can possess, since would be a subset of any other possible topology. This topology gives many properties:
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Every subset of is sequentially compact.
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Every function to a space with the indiscrete topology is continuous.
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is path connected and hence connected but is arc connected only if is uncountable or if has at most a single point. However, is both hyperconnected and ultraconnected.
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If has more than one point, it is not metrizable because it is not Hausdorff. However it is pseudometrizable with the metric .
Title | indiscrete topology |
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Canonical name | IndiscreteTopology |
Date of creation | 2013-03-22 12:48:11 |
Last modified on | 2013-03-22 12:48:11 |
Owner | mathwizard (128) |
Last modified by | mathwizard (128) |
Numerical id | 20 |
Author | mathwizard (128) |
Entry type | Definition |
Classification | msc 54-00 |
Synonym | trivial topology |
Synonym | coarse topology |