1. is the set of following elements: . By the definition, an open set in is such set, that for every element of this set there exists some open ball with centre in this element, such that this ball also belong to the set.
Thus, for any ,since , , and Let's denote with . Then the open ball with the center in x, and radius r belongs to . Hence, is an open set.
If only one of or is then we will let be either or respectively. If and simultaneously, then we can let , and solution will be the same.
2. is the set of following elements: , . By the definition,a closed set in is such set, which complement is an open set. The complement for a closed interval is:
. As we proved in 1., both of these intervals are open sets. Thus, since the union of open sets is also an open set, we obtan that is an open set. Therefore, [a, b] is a closed set.
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