# Topological Spaces on Three Points

Topological Spaces on Three Points

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:

(S,T)

S

T

S

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

∅

S

T

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

T

T

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

T

T

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.

S={1,2,3}

T

{}=∅

{1}

{2}

{3}

{1,2}

{2,3}

{1,3}

{1,2,3}

T