section13notes

# Section13notes - Lecture 13 The topology of R and metric spaces Deﬁnition neighborhood deleted neighborhood interior point boundary point closed

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Unformatted text preview: Lecture 13: The topology of R and metric spaces Deﬁnition: neighborhood, deleted neighborhood, interior point, boundary point, closed set, open set. Examples of sets which are open, closed, both, and neither. Theorem: The union of any collection of open sets is open, and the intersection of a ﬁnite collection of open sets is open. Example to show that the arbitrary union of open sets is not necessarily open. Deﬁnition: limit point, isolated point, closure of a set. Deﬁnition: open cover, compact. Heine-Borel theorem: A subset S ⊆ R is compact iﬀ it’s closed and bounded. Bolzano-Weierstrass theorem: If a bounded subset S of R contains inﬁnitely many distinct points, then there exists at least one point in R that is an limit point of S . Deﬁnition: metric, metric space. Deﬁnition: Euclidean space Rn . Examples: Euclidean metric, sup metric, and taxi-cab metric on Rn , discrete metric. Theorem: Let (X, d) be a metric space. Any neighborhood of a point in X is an open set. Deﬁnition: bounded subset of a metric space. Theorem: Let T be a compact subset of (X, d). Then T is closed and bounded, and every inﬁnite subset of T has a limit point in T . Deﬁnition: sequences and convergence in metric spaces, complete metric space. 1 ...
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## This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.

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