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Exercise 4.6. (1) Consider R² with the standard topology. Let p ER² be a point not in a closed set A. Show that inf{d(a, p) | a E A} > 0. (Recall that inf E is the greatest lower bound of a set of real numbers E.) (2) Show that R² with the standard topology is regular. (3) Find two disjoint closed subsets A and B of R² with the standard topology such that inf{d(a, b) | a E A and b e B} = 0. (4) Show that R² with the standard topology is normal.

Exercise 4.6. (1) Consider R² with the standard topology. Let p ER² be a point not in a closed set A. Show that inf{d(a, p) | a E A} > 0. (Recall that inf E is the greatest lower bound of a set of real numbers E.) (2) Show that R² with the standard topology is regular. (3) Find two disjoint closed subsets A and B of R² with the standard topology such that inf{d(a, b) | a E A and b e B} = 0. (4) Show that R² with the standard topology is normal.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Could you explain how to show 4.6 in easiest possible way(in very detail)?

Exercise 4.6. (1) Consider R² with the standard topology. Let pER² be a point not in a closed set A. Show that inf{d(a, p) | a E A} > 0. (Recall that inf E is the greatest lower bound of a set of real numbers E.) (2) Show that R² with the standard topology is regular. (3) Find two disjoint closed subsets A and B of R² with the standard topology such that inf{d(a, b) | a E A and b E B} = 0. %3D (4) Show that R² with the standard topology is normal. Definition. Let (X,J) be a topological space. (1) X is a T1-space if and only if for every pair x, y of distinct points there are open sets U,V such that U contains x but not y, and V contains y but not x. (2) X is Hausdorff, or a T2-space, if and only if for every pair x, y of distinct points there are disjoint open sets U,V such that x E U and y e V. (3) X is regular if and only if for every point x E X and closed set A C X not containing x, there are disjoint open sets U,V such that x € U and A C V. A T3-space is any space that is both T and regular. (4) X is normal if and only if for every pair of disjoint closed sets A, B in X, there are disjoint open sets U,V such that A C U and BC V. A T4-space is any space that is both T and normal.
Transcribed Image Text:Exercise 4.6. (1) Consider R² with the standard topology. Let pER² be a point not in a closed set A. Show that inf{d(a, p) | a E A} > 0. (Recall that inf E is the greatest lower bound of a set of real numbers E.) (2) Show that R² with the standard topology is regular. (3) Find two disjoint closed subsets A and B of R² with the standard topology such that inf{d(a, b) | a E A and b E B} = 0. %3D (4) Show that R² with the standard topology is normal. Definition. Let (X,J) be a topological space. (1) X is a T1-space if and only if for every pair x, y of distinct points there are open sets U,V such that U contains x but not y, and V contains y but not x. (2) X is Hausdorff, or a T2-space, if and only if for every pair x, y of distinct points there are disjoint open sets U,V such that x E U and y e V. (3) X is regular if and only if for every point x E X and closed set A C X not containing x, there are disjoint open sets U,V such that x € U and A C V. A T3-space is any space that is both T and regular. (4) X is normal if and only if for every pair of disjoint closed sets A, B in X, there are disjoint open sets U,V such that A C U and BC V. A T4-space is any space that is both T and normal.
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