(a). Prove directly that if K is a compact set in X, and x E X is any arbitrary point, then In E N such that K C Bn (x). In addition, prove using only the definition of compact set that if z E X is an accumulation point of K, then necessarily z E K. (b). Show that the intersection of any family of compact sets is compact. (c). Show that the union of a finite number of compact sets is compact. (d). Show that the union of an arbitrary collection of compact sets might not be compact