List all possible topologies on the set X={a,b}.

Solution:

When considering the set X = {a, b}, there are several possible topologies that can be defined on this set. A topology on X is a collection of subsets of X that satisfy certain conditions. Here are all possible topologies on the set X = {a, b}:

1) The discrete topology: In this topology, every subset of X is considered an open set. Therefore, the open sets are {∅, {a}, {b}, {a, b}}. This is the most fine-grained topology on X.

2) The indiscrete topology: In this topology, only the empty set and the entire set X are considered open. Therefore, the open sets are {∅, X}. This is the coarsest topology on X.

3) The trivial topology: This is a topology where only the empty set and the set X itself are considered open. Therefore, the open sets are {∅, X}.

4) The cofinite topology: In this topology, the open sets are the empty set and any subset of X whose complement in X is finite. Therefore, the open sets are {∅, {a}, {b}, X}.

In general, there can be more topologies defined on a set, but for a set with only two elements, these are the possible topologies.