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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
Theorem: Let (X, d) be a metric space. Any neighborhood of a point in X is an
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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