Open Sets

Real Analysis · Axiom Academy

The foundation of topology: sets where every point has room to breathe, preserved under arbitrary unions and finite intersections. Take a set U inside a metric space X . We call U open when every one of its points is an interior point: no matter which you pick, you can fit a whole open ball B(x,r) around it that still lies entirely inside U . Every point of an open set is interior The cleanest example is an open interval. Watch what happens to a point near the endpoint : inside (0,1) every ball can shrink to fit, but at the endpoint of [0,1] a ball always spills out of the set. Every point of (0,1) has a small interval around it that stays inside (0,1) . The point 0 belongs to [0,1] , but any interval around 0 includes negative numbers outside the set. More examples. The whole real line is open. The empty set is open — vacuously, since there is no point to test. Any open interval (a,b) is open. But closed intervals [a,b] and half-open intervals [a,b) are not open, because their included endpoints fail the ball test. The first topology axiom about combining: a union of open sets — no matter how many — is still open. The animation shows why a point x in the union always keeps a ball: it already lives inside one of the pieces, and that piece's ball works for the whole union. is open. Even the infinite union is open. 4. Finite Intersections Stay Open

This is the written version of the interactive lesson above. See the full Real Analysis course.