There are several interpretations of the original question, but one is, why focus on open sets rather than closed sets? I have an unusual answer.

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.

But as you continue with constructive topology, more advanced things fail, such as the Tychonoff Theorem (which implies the axiom of choice and thus excluded middle). Then you learn that this stuff works in locale theory, so you abandon traditional topological spaces for locales. And here the duality between open and closed is restored; a locale's frame of opens can just as well be interpreted as a coframe of closeds, and only tradition tells us to do the first. (In the locale of real numbers, the union of the closed sublocales $ [ 0 , 1 ] $ and $ [ 1 , 2 ] $ is the closed sublocale $ [ 0 , 2 ] $, and the thing that you can't prove constructively is that every point in this union belongs to at least one of its addends.)

So this answer only works in a very unusual frame of mind: setting off down an unusual path but not going all the way.

There are several interpretations of the original question, but one is, why focus on open sets rather than closed sets? I have an unusual answer.

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 near $ 1 $ is in $ [ 0 , 1 ] $ or in $ [ 1 , 2 ] $.) So open sets are better behaved and naturally you prefer to axiomatise them.

But as you continue with constructive topology, more advanced things fail, such as the Tychonoff Theorem (which implies the axiom of choice and thus excluded middle). Then you learn that this stuff works in locale theory, so you abandon traditional topological spaces for locales. And here the duality between open and closed is restored; a locale's frame of opens can just as well be interpreted as a coframe of closeds, and only tradition tells us to do the first. (In the locale of real numbers, the union of the closed sublocales $ [ 0 , 1 ] $ and $ [ 1 , 2 ] $ is the closed sublocale $ [ 0 , 2 ] $, and the thing that you can't prove constructively is that every point in this union belongs to at least one of its addends.)

So this answer only works in a very unusual frame of mind: setting off down an unusual path but not going all the way.

There are several interpretations of the original question, but one is, why focus on open sets rather than closed sets? I have an unusual answer.

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.

But as you continue with constructive topology, more advanced things fail, such as the Tychonoff Theorem (which implies the axiom of choice and thus excluded middle). Then you learn that this stuff works in locale theory, so you abandon traditional topological spaces for locales. And here the duality between open and closed is restored; a locale's frame of opens can just as well be interpreted as a coframe of closeds, and only tradition tells us to do the first.

So this answer only works in a very unusual frame of mind: setting off down an unusual path but not going all the way.

There are several interpretations of the original question, but one is, why focus on open sets rather than closed sets? I have an unusual answer.

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.

But as you continue with constructive topology, more advanced things fail, such as the Tychonoff Theorem (which implies the axiom of choice and thus excluded middle). Then you learn that this stuff works in locale theory, so you abandon traditional topological spaces for locales. And here the duality between open and closed is restored; a locale's frame of opens can just as well be interpreted as a coframe of closeds, and only tradition tells us to do the first. (In the locale of real numbers, the union of the closed sublocales $ [ 0 , 1 ] $ and $ [ 1 , 2 ] $ is the closed sublocale $ [ 0 , 2 ] $, and the thing that you can't prove constructively is that every point in this union belongs to at least one of its addends.)

So this answer only works in a very unusual frame of mind: setting off down an unusual path but not going all the way.