Finding Closure and Boundary

Topology · Axiom Academy

EXAMPLE Finding Interior, Closure, and Boundary Step-by-step worked examples in topological spaces Step 1: Find the Interior Int(A) The interior consists of all points that have an open neighborhood entirely contained in . For any point , we can find an open interval contained in . However, the point is problematic: any open interval containing 1 must extend beyond 1 (since open intervals are of the form ), and will contain points greater than 1, which are not in . Step 2: Find the Closure Cl(A) The closure is the smallest closed set containing . It consists of all points in plus all limit points . Every point in is already in . The point is a limit point because every open interval contains points from (specifically, points in ). Points outside are not limit points of . The boundary consists of points where every open neighborhood contains both points in and points not in . At : Any interval contains points in (like ) and points not in (like ). Step 1: Find the Interior Int(ℚ) For any rational number , every open interval contains both rationals AND irrationals (by the density of both sets in ). This means no point in has an open neighborhood contained entirely in . Step 2: Find the Closure Cl(ℚ) Every real number is a limit point of because rationals are dense in . For any real number and any , the interval contains rational numbers. Therefore, every is in the closure of . The set consists of the points: This sequence approaches 0 as , but 0 itself is not in the set.

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