Question

10. Let be the empty set. Let X be the real line, the entire plane, or, in the three-dimensional case, all of three-space. For each subset A of X, let X-A be the set of all points x E X such that x ¢ A. The set X-A is called the complement of A. In layman's terms, the complement of a set is everything that is outside of that set. Or you can say, the complement of a set that is everything not in the set a. Argue that is both open and closed. b. Let U be a subset of X. Show that if U is open, then X U is closed. c. Let F be a subset of X. Show that if F is closed, then X - F is open. d. Using parts b., or c, argue that X is both open and closed. (It can be shown that 0 and X are the only subsets of X that are both open and closed

please do the question and explain each parts of the question clearly. thanks

Answer 1

Using the definition of the closed set and the open set, we solved this problem. Please find below the solution.