Topological Spaces on Three Points

A topological space can be defined as a pair

(S,T)

, where

S

is a set of points and

T

(a topology) is a collection of subsets of

S

called open that satisfy four conditions:

1. The empty set

∅

and the set

S

itself belong to

T

.

2. Any finite or infinite union of members of

T

also belongs to

T

.

3. The intersection of any finite number of members of

T

also belongs to

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

S={1,2,3}

, with

T

selected from the power set of three points:

{}=∅

,

{1}

,

{2}

,

{3}

,

{1,2}

,

{2,3}

,

{1,3}

and

{1,2,3}

. The set

T

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