Closure and Interior Operations

Real Analysis · Axiom Academy

LESSON Closure and Interior Operations Every set S has a largest open core , a smallest closed hull , and a thin skin between them — watch each one get built by testing neighborhoods. 1. The Interior — the Open Core The interior is the largest open set contained in S : the points that are safely inside . A point x is an interior point when some open ball around it — an interval for a radius r>0 — fits entirely inside S . Example: the half-open interval S=(0,1] Slide an interior point and its ball stays inside, so every point of (0,1) qualifies. But at the kept endpoint x=1 , no ball fits — every (1-r,1+r) pokes past 1 into points not in S . So . 2. The Closure — Filling in the Limit Points The closure is the smallest closed set containing S — equivalently, S together with all its limit points . A point x joins the closure when S comes arbitrarily close: every ball , no matter how small, still meets S . Example: S=(0,1] presses up against 0 The points all lie in S and march toward 0 . So 0 is a limit point even though — every ball around 0 catches infinitely many of them. Closure fills it in: . The kept endpoint 1 is already in S , so it needs no filling. 3. The Boundary — the Skin Between The boundary is what closure adds but interior leaves out: . A boundary point is a point of indecision — every neighborhood of it meets both S and its complement S^ c . It is touched by the set yet never safely inside it. boundary = closure minus interior every ball straddles S and S^ c

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