1.84 We call pa cluster point of E, provided that for all e > 0 there exists e e E such that 0 < le – pl < e. (See Defintion 2.1.1.) Let ECR be any set with the property that there is a cluster point p of E such that p ¢ E. Show that there exists an open cover of E that has no finite subcover. Justify your claims. (Note: Exercise 1.83 is an example of the claim of this exercise.) Definition 2.1.1 A point a is called a cluster point of the set D if and onły if for all 8 > 0 there exists x E D such that 0 < |x – a| < 8. Thus a cluster point a of a set D has the property that it is always possible to find points x E D for which x # a and yet x is as close to a as we like.

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