I was reading a set theory book and they claim that there are as many open sets in $\mathbb R$ as real numbers (usual topology).
I tried using the base of intervals with irrational extremes but found nothing.
Any hint is welcome.
2026-03-30 05:12:04.1774847524
How many open sets of R do exist?
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As |{ ($\infty$,r ) : r in R }| = c = |R|,
c <= |topology R|.
Each open set is a countable, pairwise disjoint,
union of open intervals and rays.
How many open intervals and rays are there?
How many finte collections of open intervals and rays and are there?
How many denumberable collections of open rays and are there?
c$^{|N|}$ = c.
Add them all together and what do you get?