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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.

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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

2 Answers 2

up vote 2 down vote accepted

In each case, the property in question has a definite behaviour with respect to comparison of topologies, which you can determine by going through its definition and checking whether its conditions are easier or harder to fulfil when there are additional open sets. The set of topologies on a set is a complete lattice with least element the trivial topology and greatest element the discrete topology, so each of the questions is answered by one of these two.

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Thank you for your answer. –  blindman Oct 19 '12 at 9:22

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.

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Thank you for your answer. –  blindman Oct 19 '12 at 9:23

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