Cfrgtkky

I'm working on Question #5.v in chapter 3.1 of Sidney Morris's Toplogy without Tears.

I'm supposed to prove that $mathbb{R}$ has $2^{mathfrak{c}}$ distinct uncountable dense subsets.

I think my method gets at least halfway there.

Here's what I've done:

Note that $|mathbb{I}| = |mathbb{I}^{+}| = mathfrak{c}$

$forallalphain mathbb{I}^{+}, $ let the set $B_alpha = {mathbb{R}^{-}cupmathbb{Q}^{+}cup{alpha}}$

Then, each $B_alpha$ is distinct, uncountable, and dense in $mathbb{R}$, and there are $mathfrak{c}$ of them indexed to the positive irrationals.

Now, consider that $|P(mathbb{I}^{+})| = 2^{mathfrak{c}}$, where each element of the power set is a distinct set of positive irrationals.

Since each element of the power set can be mapped 1-1 to some union of $B_alpha$'s, we have $2^{mathfrak{c}}$ distinct uncountable dense subsets of $mathbb{R}$.

If this is correct, I have two further questions:

1) My construction identifies $2^{mathfrak{c}}$ of required type of subset. Am I done? That is, should I be trying to determine whether there are exactly $2^{mathfrak{c}}$ or more?

2) Is there a cleaner/more elegant way to construct $2^{mathfrak{c}}$ subsets of the required type?

1) My construction identifies $2^{mathfrak{c}}$ of required type of subset. Am I done? That is, should I be trying to determine whether there are exactly $2^{mathfrak{c}}$ or more?

$mathbb{R}$ has $2^{mathfrak{c}}$ subsets. So yeah, there can't be more in the sense of cardinality.

2) Is there a cleaner/more elegant way to construct $2^{mathfrak{c}}$ subsets of the required type?

By "required type" you mean "dense"? If not then obviously that totally depends on what "required type" means and it is a case-by-case study.

If yes, then your construction can be simplified: for any subset $Asubseteqmathbb{R}backslashmathbb{Q}$ the subset $mathbb{Q}cup A$ is dense. And $mathbb{Q}cup A=mathbb{Q}cup B$ if and only if $A=B$ (under the assumption that both $A,Bsubseteqmathbb{R}backslashmathbb{Q}$). Since $mathbb{R}backslashmathbb{Q}$ is of cardinality $mathfrak{c}$ then you're done.