I would say that topology is defined in terms of open sets and closed sets. I think it is motivated by two theorems from the Calculus. The Bolzano-Weierstrass theorem and the intermediate value theorem.
In its simplest form, the Bolzano-Weierstrass theorem says that an infinite subset of a closed bounded interval $[a,b]$ of the real numbers has a limit point. You find that limit point as follows. As there are infinitely many points in the set, then there are infinitely many in the left half or the right half of $[a,b]$. Say the left half. Divide that interval in half and there are infinitely many in one half or the other. Proceed this way to produce a sequence of closed intervals $I_{n+1}\subset I_n$ with the length of the $n$th interval equal to $\frac{b-a}{2^n}$. By Cantor's theorem $\cap_n I_n$ is nonempty, and the point in there is your limit point.
It didn't really make a difference if you broke the interval into two pieces or 10 pieces. This leads to the notion of compactness, by saying that every open cover has a finite subcover. The fact that the intersection of the closed intervals $I_n$ is nonempty is the complementary notion that a collection of closed subsets with the finite intersection property has nonempty intersection. The proof of the Bolazno-Weierstrass Bolzano-Weierstrass theorem leads you to think of open and closed sets.
A similar analysis of the proof of the intermediate value theorem leads likewise to open sets and closed sets.
Really, the consept concept of a topology was an incredible creative leap, that allowed people to take ideas from the Calculus and apply them in other places. Similar leaps to me, are the notion of sigma algebra, distribution (in the PDE sense), and the construction of homological algebra. :)
