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.
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