Whether there exists a Deterministic Infinite Automata (DIA), which accepts all strings in $L$ and rejects all strings not in $L$?

Question

Given an automata DIA $I = (Q,\Sigma,\delta,q_0,F)$, and the set of states $Q$ is infinite. The set of characters $\Sigma$ is still finite.

Wondering whether there is an $I$ and an arbitrary language $L$ over $\Sigma$, such that $I$ accepts all strings in $L$ and rejects all strings not in $L$?

Answer 1

Welcome to MSE!

Hint: Build a tree of all possible words in $\Sigma^*$. Put accepting states wherever you need them to go.

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I hope this helps ^_^