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topology handout

topology handout - Weierstrauss Theorem if a function f X R...

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Weierstrauss Theorem: if a function f : X ! R is continuous and X is compact, then f achieves a maximum on X: Notation (ball): A ball with radius " around a point x is denoted as B ( x; " ) : A set X is bounded if, for each x 2 X; there exists an " > 0 such that X B ( x; " ) . X is open if, for all x 2 X; there exists some " > 0 such that the open ball B ( x; " ) X: x if, for all " > 0 ; there exists some n such that x i 2 B ( x i ; " ) for all i > n: Translation: a sequence converges if it±s possible for me to keep getting closer and closer to the limit point x; and once I get close (i.e. once I get within " of x; I never move farther away. A set X is closed if no sequence of points in X converges to a point outside of X:
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