Limit Points and Accumulation

Real Analysis · Axiom Academy

LESSON Limit Points and Accumulation How points cluster together — the accumulation points that turn a set into its closure. A point p is a limit point of a set S if every neighborhood of p contains at least one point of S different from p itself. No matter how small you make the window around p , the set still has points inside it — so S has points arbitrarily close to p . Every -neighborhood meets S at a point other than p 2. Limit Points vs. Isolated Points Not every point of a set is a limit point. An isolated point of S is a point of S that has a neighborhood meeting S only at itself — there is breathing room around it with nothing else of S inside. A point of S is either a limit point or isolated; it cannot be both. Every neighborhood, however small, catches another point of S . Example: every point of (0,1) — neighbors crowd in from both sides. Some neighborhood contains no other point of S . Example: 5 in — alone in a clear window. Consider . As n grows the points march toward 0 , getting arbitrarily close but never landing on it. Watch where they accumulate. What accumulates, and what is isolated Every point 1/n is isolated : the gap to its nearest neighbor leaves a clear window around it. The point 0 is a limit point — for every some 1/n lies in — yet . No other point is a limit point, so the derived set is exactly . 4. The Derived Set and the Closure

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