Thread: Topology Help

Find 3 topologies on thefive point set X = {a,b,c,d,e} such that the first is finer then the second and the second is finer then the third, without using either the trivial or discrete topology.

Hello,

Originally Posted by r2dee6

Find 3 topologies on thefive point set X = {a,b,c,d,e} such that the first is finer then the second and the second is finer then the third, without using either the trivial or discrete topology.

A topology A is said to be finer than a topology B, if any open set of B is an open set of A.

Knowing that, it's quite easy
Consider the less fine topology, let's say {0,{a},X} (0 designs the empty set)

Then you have to find a topology that contains {a} (it'll always contain 0 and X, because it's the definition) and more sets.
So you can add {b} and {a,b}. You have to add {a,b} because the union of two elements of a topology is in the topology and {a} U {b} = {a,b}
Hence {0,{a},{b},{a,b},X} can be the second finest topology.

You need to find a third one, that contains {a},{b},{a,b}
Let's add {c}
Since it has to be stable by union (for intersection, it's trivial because it's the empty set), you have to add {a} U {c}={a,c}, {b} U {c}={b,c} and {a,b} U {c}={a,b,c}
So the finest topology can be {0,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c},X}