Give an example for each of the following:
A set having no limit point.
Definition. A limit point (accumulation point) of a set in a metric space is a point any puncted neighborhood of which contains an element of . The set of all limit points of is denoted by .
In other words, is the set of all points at which the set is locally infinite. Really, if there exists a neighborhood of such that is finite then, by putting , we have that a puncted open ball contains no elements of . This means that is not a limit point of . Conversely, if any neighborhood of contains an infinite subset of then a puncted neighborhood of also contains an infinite subset of , hence, is a limit point of .
Examples of locally finite sets (i.e. sets having no limit point):
1) any finite set;
2) the set of integers (any open interval of length contains or less integers);
3) the set in the metric space with the standard metric. Indeed, if then the interval is a neighborhood of such that the puncted contains no elements of . Therefore, . If and then put .We have , thus, . The interval is an open neighborhood of containing no elements of . Thus, and therefore, .
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