I want to prove the following simple lemma:


Let $T(x)=2x\mod 1$, be defined on $[0\,,1)$ to itself,suppose for any $k\ge 0$, $T^{k}(a)\notin(a\,,b)$, then we have 


$\Omega=\{x\in[0\,,1):\mbox{for any k}\geq 0$, $T^{k}(x)\notin(a\,,b)\}$ is not closed. where $(a\,,b)\subset[0\,,1)$. 



Open dynamical system of doubling map
I thought about my previous comment and I can assure that in general your lemma is false. If $a > \frac{1}{2}$ and you consider the hole $(a,1)$, $(\Omega, T\mid\_{\Omega})$ is conjugated to a $\beta$ shift see J. Nilsson. On numbers badly approximable by dyadic rationals. Israel J. Math., 171:93-110, 2009. Also, Theorem 3.5 of S. Bundfuss, T. Kruger, and S. Troubetzkoy. Topological and symbolic dynamics for hyperbolic systems with holes. Ergodic Theory Dynam. Systems, 31(5):1305-1323, 2011, gives a negative answer to your question. Here, the authors do not consider $T$ as a continuous map. 


