# Thread: Density of a Set

1. ## Density of a Set

Let $Y\subset X$ and let $A\subset X$. If $A$ is countable, $X$ is complete, and $Y\cup A=X$, is $Y$ necessarily dense in $X$?

Prove it or give a counterexample.

If the answer is no, then does making $X$ compact change anything?

2. Originally Posted by redsoxfan325
Let $Y\subset X$ and let $A\subset X$. If $A$ is countable, $X$ is complete, and $Y\cup A=X$, is $Y$ necessarily dense in $X$?

Prove it or give a counterexample.

If the answer is no, then does making $X$ compact change anything?
Kind of an obvious counterexample, isn't there? Take A and Y to be completely disjoint!

3. Originally Posted by HallsofIvy
Kind of an obvious counterexample, isn't there? Take A and Y to be completely disjoint!
But $X$ is complete. I feel like that can't happen if $Y$ and $A$ are disjoint.

4. $X = \left[ { - 1,0} \right] \cup \left\{ {n^{ - 1} :n \in \mathbb{Z}^ + } \right\}$?

5. Originally Posted by Plato
$X = \left[ { - 1,0} \right] \cup \left\{ {n^{ - 1} :n \in \mathbb{Z}^ + } \right\}$?
I think that works. It would be complete because $Y$ is complete and the only Cauchy sequences in $A$ are all the same number or the sequence $\{a_n\}=\frac{1}{n}$ and both of those converge to something in $X$.

It also answers the second part of the question because this set is also compact.