How is the name "open set" in topological space consistent with the intuitive definition of an open set? I can accept that "open set" is just a name used for the three characteristics of a topology in the definition.
Usually, when reading about open sets the definition goes like DEF := "an open set is a set that only includes its interior and not its boundary".
So, some questions:
a) Is this definition DEF just only true for metric spaces and does not have any correspondence to topological spaces?
b) If it does correspond to topological spaces, then how? For example, wikipedia has this image to describe what is a topology and what not: https://upload.wikimedia.org/wikipedia/commons/f/ff/Topological_space_examples.svg I do not even understand what the boundary or interior of 2 should include, what would that be?
c) If DEF does not hold for topological spaces, is "closed set" really just a name that has no further characteristics whatsoever?

For a topological space, your intuitive "definition" is true, since for any set $A$ we have $\partial A = \bar{A}-A^o$. If $A$ is open then $A = A^o$ and so $\partial A\cap A = \emptyset$. Conversely, if the last equality holds then $A^o = A-\partial A = A$.
Only problem is, the definition of the boundary already relies on the definition of an open set; so this definition for an open set is circular. In a metric space, a set is open if each of its elements have an $\epsilon$-ball around it contained in the set. Intuitively, these sets do not contain "boundary points", since any ball around a boundary point would contain points outside of the set. Then it turns out that these open  sets are closed under finite intersection and arbitrary union, and the whole space and $\emptyset$ are open. These last four requirements are then taken to be the only necessary properties of an open set in a topological space. 
