need all steps and complete work solution please

10. 11. 12. A complex number 2 is an interior point of a set S of complex numbers if there is a neighborhood of 2 that contains only points in 5, whereas w is a boundary point of S if each neighborhood of w contains at least one point in S and one point not in S. Prove the following: (i) A set of complex numbers is open if and only if each point in S is an interior point of S. (ii) A set of complex numbers is open if and only if it contains none of its boundary points. (iii) A set of complex numbers is closed if and only if it contains all its boundary points. Let D = [z E C | |z| S 1} be the closed unit disk and let S be a subset of D that includes the interior of the disk but is missing at least one point on the bounding circle of the disk. Show that S is not a closed set. Prove that a set of complex numbers is closed if and only if it contains all its accumulation points. (See Problem 6 for the denition of an accumulation point.) Prove that a set consisting of nitely many complex numbers is a closed set in C. (Hint: Show that a nite set has no accumulation points.)