Subject GRE-Style: Compactness Arguments

GRE Math Subject · Axiom Academy

EXAMPLE Subject GRE-Style: Compactness Arguments 3 worked problems on compactness, Heine-Borel, and continuous functions on compact sets Key Facts: Heine-Borel: A subset of is compact iff it is closed and bounded. A continuous function on a compact set is bounded and attains its bounds (Extreme Value Theorem). A continuous function on a compact set is uniformly continuous. Question: Which of the following subsets of is compact? (A) : Bounded but not closed (0 and 1 are limit points not in the set). Not compact. (B) : Closed but not bounded. Not compact. (C) : Bounded but not closed ( is a limit point not in the set). Not compact. (D) : Bounded but not closed (0 is a limit point not in the set). Not compact. (E) : Bounded (contained in ) and closed (the only limit point, 0, is included). Compact! GRE Tip: The set without 0 is the classic "almost compact" trap. Always check whether limit points are included. Continuous Image of a Compact Set Question: Let be continuous. Which of the following must be true? (A) I only (B) I and II (C) I and III (D) II and III (E) All of I, II, III I. Bounded: A continuous function on a compact set is bounded. This is part of the Extreme Value Theorem. TRUE. II. Attains maximum: The Extreme Value Theorem guarantees that attains both its supremum and infimum on . TRUE. III. Uniformly continuous: By the Heine-Cantor Theorem, a continuous function on a compact set is uniformly continuous. TRUE.

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