2Show that N is not compact by describing an open cover for

2.Show that N is not compact by describing an open cover for N that has no finite sub-cover (that is, using just the definition, and not Heine–Borel theorem).

Solution

Given N

and N is not compact

Let G = {Bn}

where Bn = (n, n) for n N.

Then G is an open cover of N which has no nite sub cover