Why the definition of topology is what it is? I recently attended an interview for PhD. One of panel members asked me definition of topology which I answered and next question was why the definion supposed to be that way. 
   I explained through an example. Is it right to explain definition using example? What could be a better answer?
Any help will be deeply acknowledged.
I don't think one example is enough for justifying such a definition. If you only have one specimen it's premature generalization to introduce a classification of that specimen.

As for the set of properties that you use for classification it will depend on which properties that are actually used in proving corresponding propositions for different spaces.

Also note that for a classification to be meaningful you need to have examples that addresses specimina that lies outside subclasses. For example for topological spaces you need at least one example for a space that is not metric (ie if all your examples are metric then the definition of metric space would suffice).

In addition the examples need to be of general importance (for the definition is to be of general interrest).
Consider real line with usual metric, then open sets are union of open intervals. 
we have following results:


Any union of open sets is open
Finite intersection of open sets is open
Empty set and real line are open


Now,  we can ask a question,  For any set X,  and a collection of subsets of X satisfying the above results.  If it satisfy,  is called a topology on X