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2. Define the concepts of (a) first countable space; (b) second countable space; (c) separable space and a (d) Lindelof space.

2. Define the concepts of (a) first countable space; (b) second countable space; (c) separable space and a (d) Lindelof space.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...
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2. Define the concepts of (a) first countable space; (b) second countable space; (c) separable space and a (d) Lindelof space. (i) Prove that every second countable space is separable; (ii) Prove that a separable metric space is second countable; (iii) Prove that a compact space is Lindelof; (iv) Give an example of a Lindelof space which is not compact;
Transcribed Image Text:2. Define the concepts of (a) first countable space; (b) second countable space; (c) separable space and a (d) Lindelof space. (i) Prove that every second countable space is separable; (ii) Prove that a separable metric space is second countable; (iii) Prove that a compact space is Lindelof; (iv) Give an example of a Lindelof space which is not compact;
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