Closed Sets

Real Analysis · Axiom Academy

A set is closed when it contains all its limit points — equivalently, when its complement is open. We unpack both views and the dual topology axioms. A point x is a limit point of F if every open ball contains a point of F other than x — equivalently, some sequence with . The set F is closed when it owns all of them: if a sequence stays in F , its limit cannot leak out. F is closed: it contains all its limit points Sequential form: a trapped sequence keeps its limit inside Here is the definition most textbooks lead with: F is closed precisely when its complement is open . The two definitions agree — if F owns its limit points, then any point outside F has breathing room (a ball missing F ), so is open, and conversely. Take . Since F contains all its limit points, x is not one — so some ball avoids F entirely, i.e. . Every point of has such a ball, which is the definition of open. For F=[0,1] , the complement is a union of open rays — visibly open. Closed sets satisfy the exact dual of the open-set axioms (apply De Morgan to the open axioms and complement). Two of the three are unconditional: and the whole space X are both closed (their complements X and are open). Any intersection of closed sets — even infinitely many — is closed. A finite union of closed sets is closed (the infinite case is the catch in Step 4). Closed sets mirror open sets: is unrestricted, is restricted — the reverse of the open-set rules.

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