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Topological Spaces on Three Points

A topological space can be defined as a pair

(S,T)

, where

is a set of points and

(a topology) is a collection of subsets of

called open that satisfy four conditions:

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

S={1,2,3}

, with

selected from the power set of three points:

{}=∅

{1}

{2}

{3}

{1,2}

{2,3}

{1,3}

and

{1,2,3}

. The set

is a topological space only if the three conditions listed are satisfied.

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