Metric Spaces — Open Sets, Closed Sets, Compactness

GRE Math Subject · Axiom Academy

LESSON Metric Spaces, Open/Closed Sets, Compactness The topological backbone of analysis — definitions, examples, and the Heine–Borel theorem Step 1: Metric Spaces — Definition and Examples Definition: A metric space is a pair where is a set and is a function satisfying, for all : Positivity: , with equality iff Standard examples you must know: The discrete metric on any set: (continuous functions on ) with Open set: A set is open if for every , there exists such that . Closed set: A set is closed if its complement is open. Arbitrary unions of open sets are open; finite intersections of open sets are open Arbitrary intersections of closed sets are closed; finite unions of closed sets are closed and are both open and closed ("clopen") A set can be neither open nor closed (e.g., in ) Interior: — the largest open set contained in Closure: — the smallest closed set containing Example: In , let . Then , , . Step 3: Limit Points and Sequential Characterizations Limit point: is a limit point (accumulation point) of if every open ball around contains a point of other than itself. Equivalently: there is a sequence in (distinct from ) converging to . Sequential characterizations (crucial for GRE): A set is closed iff it contains all its limit points iff whenever with each , we have . A point is in iff there exists a sequence in converging to . Definition (open cover): A collection of open sets is an open cover of if .

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