# Number of subsets satisfying a topological property of a topological space

Let $X$ be a nonempty set. Find the topology $\tau$ on $X$ satisfying one of the following conditions:

• $(X, \tau)$ has the largest number of compact subsets.

• $(X, \tau)$ has the least number of compact subsets.

• $(X, \tau)$ has the largest number of connected subsets.

• $(X, \tau)$ has the least number of connected subsets.

• $(X, \tau)$ has the largest number of dense subsets.

• $(X, \tau)$ has the least number of dense subsets.

I would like to thank all for their helping and comments.

-
Hint: What subsets are dense/compact/connected when $X$ is endowed with the discrete topology or with the trivial topology? – martini Oct 19 '12 at 8:54

Try thinking about when $X$ is given the discrete topology, and see if any of them become trivial. For example, if $X$ is discrete, then every subset is dense. Then you should be able to use arguments involving the discrete and trivial topologies to answer the other questions.