# On the connectedness of a set relating to prime numbers

Suppose $$r:Bbb Nto (0,1)$$ is a function given by $$r(n)$$ is obtained by putting a point at the beginning of $$n$$ instance $$r(34880)=0.34880$$ and let $$N_1:={2n-1mid ninBbb N}$$ and $$lt_1$$ be a total order relation on $$N_1$$ with: $$forall m,ninBbb N,,2n-1lt_12m-1LeftarrowRightarrow r(2n-1)lt r(2m-1)$$

then $$N_1$$ is a Hausdorff space induced by $$lt_1$$.

Theorem: $${2p-1mid pinBbb P}$$ is dense in $$N_1$$. proof

and assume $$J:={2p_{i+1}+2p_{i+2}-1mid p_i$$ is $$i$$_th prime number$$}$$.

Question: Is $$N_1setminusbar J$$ connected?

Thanks a lots!