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!

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