In Munkres' Topology, the axiom of choice is defined as follows:
Given a collection $A$ of disjoint nonempty sets, there exists a set $C$ containing exactly element from each element of $A$; that is a set $C$ such that $C$ is contained in the union of elements of $A$, and for each $a$ in $A$, the intersection of $C$ and $a$ is a singleton set.
Now the text asks to prove the existence of a choice function for an arbitrary collection of non-empty sets. Here's what I did:
For each $a$ in $A$, consider the set $\{a\}$. The axiom of choice as defined above guarantees that we can pick out the so called representative element of the set $a$. Call it $c_a(a)$. We can do this for every $a$ in $A$. So now, we define our choice function $c$ such that for any $a$ in $A$, $$c(a)=c_a(a)$$ This defines the choice function completely and completes the proof.
Is this proof correct, because the proof given in the book is something different and made me wonder if this proof wasn't sophisticated enough. Thanks a lot for your time.
No, it's not correct.
The formulation of the axiom by Munkres requires the sets to be disjoint, you didn't not define any disjoint sets, or perhaps you have, the $\{a\}$'s, but then you get a set $C$ which chooses from each singleton an element. Not from each set in your family of nonempty sets.