What happens when ε-clos(X) contains duplicate elements? Example: (1+0)*1 I am learning about state machines from the Stanford online cs143 course. Here I have tried to build a nondeterministic finite automaton for the regular expression (1+0)*1, but when I attempt to convert it to a DFA I find that the epsilon closure of the initial state A ends up having two outputs for the value 1, which I imagine would be impossible for a deterministic automaton.

I understand that my NFA differs from the instructors, but I'm unsure of whether or not it is valid. If it is valid, why am I having difficulty converting it to a DFA?
Yes, your NFA is correct. You are probably making some mistake during your conversion to a DFA.

Note that after the conversion from the NFA to the DFA, the states will no longer be $S = \{A,B,C,D,E\}$. Instead they will be all possible subsets of $S$. So in your case $A$ is not giving two different transitions to $C$ and $E$, instead it is giving one single transition to the state $\{C,E\}$