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 p E R2 be a point not in a closed set A. Show that ind(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 subsetsA and B of R2 with the standard topology such that ind(a,b) | aEAandb EB} =0. (4) Show that R2 with the standard topology is normal. Denition. Let (X , 5") be a topological space. (1) X is a Tl-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 "Pg-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 setA C X not containing )6, there are disjoint open sets U, V such that x E U andA C V. A T3space is any space that is both T1 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 B C V. A Tat-space is any space that is both T1 and normal