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 ina closed set A. Show that inf{d(a, p) | a E A} > 0. (Recall that inf E is the greatestlower 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 thatinf{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 setsU,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 pointsthere 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 containingx, there are disjoint open sets U,V such that x € U and A C V. A T3-space is anyspace 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 aredisjoint open sets U,V such that A C U and BC V. A T4-space is any space thatis both T and normal.

part(a) Let A be a closed set in ℝ2. Let a point p∉A⇒p∈Ac As, we know A is a closed set thus Ac is…

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.