Open and Closed Sets

Intro to Proofs · Axiom Academy

The four rules that govern how open and closed sets behave under unions and intersections — and the one place where "infinite" breaks everything. A set U is open if every point of U has a little breathing room: an open ball B(x,r) around it that still lies entirely inside U . A set C is closed if its complement is open — equivalently, C contains all of its boundary points. Open: room to move around every point Closed: the complement is open Take any collection of open sets — two, a thousand, or infinitely many — and union them together. The result is always open. Watch overlapping open intervals merge into one seamless open set below. Why it works: pick any point x in the union. Then x lands in at least one of the sets, say . That is open, so it already hands us a ball . One set is all we ever needed — so the count of sets is irrelevant. 3. Finite Intersections — and Where "Finite" Bites Intersecting finitely many open sets keeps you open: each set gives a radius r_i , and the smallest one, , still fits inside all of them. But that minimum only exists because the list is finite. Drop "finite" and the argument — and the conclusion — can fail. The classic counterexample. Consider the open intervals for Each is open, and they are nested, shrinking toward 0 . Their infinite intersection contains exactly the points inside every interval — and the only such point is 0 . Watch the intervals collapse: 4. Closed Sets: The Mirror Image

This is the written version of the interactive lesson above. See the full Intro to Proofs course.