TITLE: Natural non-trivial topology on ${\mathbb R}$ such that there are more than $2^{\mathbb N}$ open sets
QUESTION [5 upvotes]: The collection of open sets of ${\mathbb R}$ with standard topology only has cardinality $2^{\mathbb N}$ because we can map the collection of countable sets of intervals with rational endpoints onto the collection of open sets. Of course if we declared all subsets of ${\mathbb R}$ to be open, then the cardinality of the collection of open sets would higher. But is there a "natural" (e.g. interesting folklore) example of a non-trivial topology on ${\mathbb{R}}$ such that the number of open sets is more than $2^{\mathbb N}$?

REPLY [3 votes]: A good candidate is the so-called density topology, which is actually quite useful in potential theory. By definition, a set $A\subset\mathbb R$ is open for this topology iff $A$ is Lebesgue-measurable and has density $1$ at each point $x\in A$, i.e.
$$\forall x\in A\;:\; \lim_{h\to 0^+}\,\frac{m(A\cap [x-h,x+h])}{2h}=1\, , $$
where $m$ is Lebesgue measure.
One property of this topology (call it $\tau$) is that all measurable subsets of $\mathbb R$ with Lebesgue measure $0$ are $\tau$-closed. So there are $2^{\mathbf c}$ closed sets wrt to $\tau$ (and hence $2^{\bf c}$ open sets).
If you are interested, you can have a look at the following paper and the references therein: http://msp.org/pjm/1976/62-1/pjm-v62-n1-p25-p.pdf