open set (axioms)
In metric spaces “open” has a vivid meaning: a set is open if you can stand on any of its points and still have a tiny ball entirely inside the set — no point sits on the very edge. In abstract topology we keep that intuition but drop the balls. Open sets become the primitive, undefined notion, and we simply declare which sets are open by listing the rules they must collectively obey.
Those rules — the open set axioms — are: the empty set and the whole space are open; the union of any family of open sets (no matter how large) is open; and the intersection of finitely many open sets is open. A collection satisfying these is a topology, and its members are by definition the open sets.
It is worth stressing what is NOT assumed. A single point need not be open or closed; a set can be neither open nor closed (a half-open interval [0, 1) in the real line is the classic case); and a set can be both open and closed (the empty set and the whole space always are). “Open” is not the opposite of “closed” — they are independent properties tied together only by complementation.
Why “finite” intersections only: each set (-1/n, 1/n) is open in the real line, but their intersection over all n is {0}, a single point that is not open. Infinitely many open sets can intersect to something that has lost all wiggle room.
Infinite intersections of open sets can fail to be open — this is why only finite intersections are required.