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 ∈ 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 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 ∅ and X are the only subsets of X that are both open and closed.
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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 ∈ 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 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 ∅ and X are the only subsets of X that are both open and closed.
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