Can this Theorem be disproved?

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Well, if we want to disprove a statement like, "Any set of numbers that is not dense is countable," it's easy to form a counterexample by making a hole in the reals. Consider, say, the set of reals that aren't on the interval [0,1]. It's uncountable, but not dense on the entire number line.

Maybe that was kind of a dirty trick, and you want an uncountable set that's not dense on its closure. Then a better example would be the Cantor set, which is uncountable, but not dense on any (nontrivial) subset of the reals.

This set can be constructed recursively by taking the interval [0,1], taking out the middle third so that the set [0,1/3]U[2/3,1], then taking the open "middle third" of the set out of each of these to give
[0,1/9]U[2/9,1/3]U[2/3,7/9]U[8/9,1].
Then repeating the process indefinitely with every interval gives you the Cantor set.

Proof sketch of uncountability: Say you want to represent some arbitrary real number on the interval (0,1) in binary. There's a unique number in the Cantor set corresponding to any arbitrary binary number. One mapping is as follows:
If the first digit is a 0, the corresponding point in the Cantor set is in the lower third of [0,1].
If the first digit is a 1, the corresponding point in the Cantor set is in the upper third of [0,1].
Find the next digit by repeating for each subinterval. Every infinite string of binary digits then corresponds to a unique point, and vice versa. Therefore it has the same cardinality as the set of reals.

Proof sketch that it's nowhere dense:
-The Cantor set's complement is a union of open sets, therefore it's open.
-This means the Cantor set is closed, therefore it's its own closure.
-The Cantor set contains no interior points because any finite interval will not be fully covered after some finite number of iterations in constructing the set, so no intervals are contained.
-If a set's closure has no interior points, then the set is nowhere dense.

So a set can be nowhere dense, and countable, or it can be everywhere dense, and uncountable (like the rationals). It would seem these properties are logically orthogonal.

If you really want citations, better information on the Cantor set is pretty easy to find online, and it often comes up in books on fractals, set theory, topology, etc.

I don't have any at hand, save for a handout on non-hausdorff metric spaces from an analysis class, unfortunately.

But fortunately, Escuerd saved me with a much simpler example

The Real Line is Hausdorff, not to mention connected. The space you were talking about, the Real Line plus a "ghost zero" point is not connected.The Cantor Set is uncountable and it is also totally disconnected. So uncountability does not automatically imply connectedness.

I guess I didn't do it right. Firstly I shouldn't have said metric space - I was thinking more topological spaces sense (in retrospect, non-hausdorff metric space doesn't make any sense at all! )

The ghost zero example is still a topological space - disconnected, non hausdorff, perhaps, but a topological space - and every topological space is dense in itself, regardless of connectivity.

I think Pi is actually infinite. How many digits has it been calculated to? 2 million?

Over a billion, actually. And pi is finite, but irrational. The number of digits is infinite, but that doesn't mean the number itself is.