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I have this NFA:

Image of NFA

I want to simulate it with a DFA, thus I have the states {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}. Usually I look at every state and find out in which state I could end up when selecting a letter. However here the problem starts at {1}, where the dfa wouldn't have any connection for letter a and thus wouldn't be a proper DFA. How can I resolve this ?

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You missed state $\emptyset$. The transition from state $1$ when character $a$ is read goes to $\emptyset$.

Since all transition from $\emptyset$ also go to $\emptyset$, you can also use the standard convention of omitting $\emptyset$ (and the transition for character $a$ from state $1$).

If a DFA has some missing transitions, it can always be completed by first adding a non-final state $q$, and then adding all missing transitions to $q$.

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