the product space $\mathbb{R}^I$,where I denote $[0,1]$, has a countable dense subset.
6523
Prove that the space has a countable dense subset
132382
Showing a set is nowhere dense in $C([0,1])$
149317
An open subset of $\mathbb{R}$ is an at-most-countable union of disjoint open intervals
102485
Showing that a topological space is ${\rm T}_1$
70413
A vector space over $R$ is not a countable union of proper subspaces
21892
Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
138768
Showing that the family of Borel sets is countably generated
121626
Every open cover of the real numbers has a countable subcover (Lindelöf's lemma)
78082
Dense subset of two Banach spaces also dense in the intersection
142577
Algebraic proof that a set generated by irrational rotations is dense in $S^1$.
105358
In a second countable space any base has a countable subset that is still a base
144246
Subset of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$
16649
Topological space with countable open cover $\{U_\alpha\} $ with each $U_\alpha$ second-countable, is second countable
32774
Finite Cartesian Product of Countable sets is countable?
158582
Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$
108305
How can we show that the dyadic rationals are dense in $\mathbb R$?
68490
For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$