Baire's theorem says that intersection of open sets that are dense is still dense. For me, the first dense set I'm thinking to in $\mathbb R$ is $\mathbb Q$ or $\mathbb R\backslash \mathbb Q$ but no of those sets are open (and Baire's theorem doesn't apply for those set). Could someone give me an example of open set $\mathcal O\subset \mathbb R$ that is dense in $\mathbb R$ and such that $\mathcal O\neq \mathbb R$ ?
Example of open $O\subset \mathbb R$ dense s.t. $O\neq \mathbb R$
real-analysis
Related Solutions
If $A$ is a dense subset of $\mathbb{R}$, then $A\neq \emptyset$. In particular, there exists an element $a\in A$. If $B=A\setminus \{a\}$, then is $B$ dense in $\mathbb{R}$? (Hint: if there is an open subset $U$ of $\mathbb{R}$ such that $U\cap A=\{a\}$, then $A$ is not dense in $\mathbb{R}$.)
Exercise 1: Prove or give a counterexample: a finite intersection of dense subsets of $\mathbb{R}$ is dense in $\mathbb{R}$.
Exercise 2: Prove or give a counterexample: if $\{A_i\}_{i\in I}$ ($I$ is an index set) is an infinite collection of dense subsets of $\mathbb{R}$ such that the intersection of any finite number of $A_i$'s is again dense in $\mathbb{R}$, then the intersection of all the $A_i$'s dense in $\mathbb{R}$. (If you wish to view a hint, hover your cursor over the grey region directly below:
(Hint: if $x\in \mathbb{Q}$, let $A_x=\mathbb{Q}\setminus \{x\}$; prove that the intersection of any finite number of $A_x$'s is dense in $\mathbb{R}$ and determine the intersection $\bigcap_{x\in\mathbb{Q}} A_x$.)
Exercise 3: If $A$ is dense in $B$ and if $B$ is dense in $\mathbb{R}$, $A\subseteq B\subseteq \mathbb{R}$, then is $A$ dense in $\mathbb{R}$?
Exercise 4 (if you are familiar with measure theory): Let $\epsilon>0$. Prove that there exists an open dense subset $U$ of $\mathbb{R}$ such that the Lebesgue measure of $U$ is at most $\epsilon$. (If you wish to view a hint, hover your cursor over the grey region directly below:
(Hint: enumerate $\mathbb{Q}$ as $q_1,q_2,\dots$. If $n\in\mathbb{N}$, let $I_n=(q_n-\frac{\epsilon}{2^{n+1}},q_n+\frac{\epsilon}{2^{n+1}})$; prove that the union $\bigcup_{n\in\mathbb{N}} I_n$ is an open dense subset of $\mathbb{R}$ and determine the Lebesgue measure of $\bigcup_{n\in\mathbb{N}} I_n$.)
Exercise 5 (if you are familiar with path-connected topological spaces): If $U$ is an open path-connected subspace of $\mathbb{R}$ and if $U$ is dense in $\mathbb{R}$, then prove that $U=\mathbb{R}$.
I hope this helps!
$A = \mathbb{R} \setminus \{0\}$ works for this purpose, and isn't equal to $\mathbb{R}$. But that's still fairly trivial.
For a less trivial example, fix an enumeration $\{r_n\}_{n = 0}^{\infty}$ of rational numbers and a positive number $\epsilon$. Define open intervals
$$\mathcal{O}_n = \left(r_n - \frac{\epsilon}{2^{n + 2}}, r_n + \frac{\epsilon}{2^{n + 2}}\right)$$
and define $\mathcal{O} = \bigcup_n \mathcal{O}_n$. Then $\mathcal{O}$ is an open, dense subset of $\mathbb{R}$ with Lebesgue measure at most $\epsilon$.
In fact, we could (by dilating one of our intervals) make the measure of $\mathcal{O}$ equal to any given positive number $\epsilon$.
Best Answer
$$\mathbf R\smallsetminus\mathbf Z=\bigcup_{n\in\mathbf Z}(n, n+1) $$ is another example.