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4.3 Give an example to show that the arbitrary intersection of open sets is not an open set.

4.3 Give an example to show that the arbitrary intersection of open sets is not an open set.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 1TFE: Label each of the following statements as either true or false. The Well-Ordering Theorem implies...
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4.3 Give an example to show that the arbitrary intersection of open sets is not an open set. 4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = Ø. 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains some nonempty open set.
Transcribed Image Text:4.3 Give an example to show that the arbitrary intersection of open sets is not an open set. 4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = Ø. 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains some nonempty open set.
2.2 Given (R", d) with d : R" × R" → R defined by n d(х, у) — (Lk – Yk)². = - k=1 Determine whether d satisfies the triangle inequality or not. [ x = (x1, F2, . .. , In) and y = (y1, 92, ... , Yn) are elements of R"-] 2.3 Let (S, d) be a metric space and suppose that p: S x S → R is defined by d(x, y) 1+ d(x, y) p(x, y) for all points x, y E S. 2.3.1 Determine whether p satisfies the triangle inequality or not. 2.3.2 Show that (S, p) is bounded and that p(x, y) < d(x, y) for all x, y E S.
Transcribed Image Text:2.2 Given (R", d) with d : R" × R" → R defined by n d(х, у) — (Lk – Yk)². = - k=1 Determine whether d satisfies the triangle inequality or not. [ x = (x1, F2, . .. , In) and y = (y1, 92, ... , Yn) are elements of R"-] 2.3 Let (S, d) be a metric space and suppose that p: S x S → R is defined by d(x, y) 1+ d(x, y) p(x, y) for all points x, y E S. 2.3.1 Determine whether p satisfies the triangle inequality or not. 2.3.2 Show that (S, p) is bounded and that p(x, y) < d(x, y) for all x, y E S.
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