Suppose you want to do constructive mathematics. (Don't ask me why, you just do.) So you abstract the properties of open and closed subsets from the real line. Then you see that open subsets are closed under arbitrary union but only finitary intersection, OK. Dually, you see that closed sets are closed under arbitrary intersection but … not under finitary union! For example, the union of $[ 0 , 1 ]$ and $[ 1 , 2 ]$ cannot be proved to be closed. (The closure of the union is $[ 0 , 2 ]$, but to prove that the union itself is all of $[ 0 , 2 ]$ requires the lesser limited principle of omniscience. Or less formally, there is no definite method to decide whether a number is near $1$ is in $[ 0 , 1 ]$ or in $[ 1 , 2 ]$.) So open sets are better behaved and naturally you prefer to axiomatise them.