Give an example of an open cover of the set 6 8 that has no

Give an example of an open cover of the set (6 ,8] that has no finite subcover

Solution

Let F = {An} where An = (0, 8 1/ n ), for n N. Then F is an open cover for [6, 8), which has no finite subcover.

Let An = ( 1/ n , 8 1/ n ), n IN. Clearly, (0, 8) _n=3An. But there exists no finite subcover. For, if (0, 8) ^N_n=3 An for any positive integer N > 3, then we have (0, 8) ( 1/ N , 1 1/ N ). But, for any n > N we have 1 n (0, 8) and 8/ n / ( 1/ N , 1 1 N ). Hence, {Gn, n = 3, 4 · · · N} cannot be an cover for (6, 8] for any N.