I have the following question:

1. Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$.

2. If the answer in general is no, are there conditions over $T$ to do this?.

3. I'm interested to know if given a topological space $T$ and a topology on $2^T$ that is induced by the topology on $T$ one can know some topological properties of $2^T$ knowing that of $T$.