Consider the sets 13 and NN Show that each subset of RR is

Consider the sets [1,3) and [NN] .

Show that each subset of [RR] is not compact by describing an open cover for it that has no finite subcover.

Solution

I edited your question because you should make separate posts for separate questions. However, the idea behind all of them is so similar that if I do the first two you might see how to do the others.

Let [S_n=(0,3-1/n)] for [n=1,2,3...]

As [n] gets larger, [3-1/n] gets arbitrarily close to 3 (but never equals 3), so [uuu S_n=(0,3),] which covers [[1,3).] However, if we take a finite subset of this cover, let [N] be the largest number such that [S_N] is in this finite subset. No matter how large [N] is, [3-1/N<3-1/(2N)<3,] so there will be elements of [[1,3)] missing, in particular [3-1/(2N).] Thus we have found a cover such that no finite subset covers [[1,3),] so [[1,3)] isn\'t compact.

Now for the set [NN] , simply let [S_n=(-1,n),] so again

[uuuS_n=(-1,oo),] covering [NN.] But again, if [N] is the largest number such that [S_N] is in a finite subset of this union, then clearly any natural number greater than or equal to [N] isn\'t covered, so [NN] isn\'t compact.