Topological Spaces on Three Points

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A topological space can be defined as a pair , where is a set of points and (a topology) is a collection of subsets of called open that satisfy four conditions:

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1. The empty set and the set itself belong to .

2. Any finite or infinite union of members of also belongs to .

3. The intersection of any finite number of members of also belongs to .

Topological spaces are, of course, usually associated with infinite sets of points. But it is amusing to apply topology to a finite set of points. This Demonstration considers a space , with selected from the power set of three points: , , , , , , and . The set is a topological space only if the three conditions listed are satisfied.

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Contributed by: S. M. Blinder (May 2018)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshot 1: fulfills the condition for a topology

Snapshot 2: is not a topology, since the subset is missing

Snapshot 3: is not a topology, since the element is missing

Reference

[1] Wikipedia. "Topological Space." (May 11, 2018) en.wikipedia.org/wiki/Topological_space.



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