Question

The component set of the component set of the given set is not itself.​

Answer 1

Answer:

In the case of (d) NN is connected, since suppose we had a separation into two nonempty open disjoint subsets: N=U∪VN=U∪V. We know U=N−{n1,...,nk}U=N−{n1,...,nk} and that V=N−{m1,...,ml}V=N−{m1,...,ml}. Now UU and VV can't be disjoint, as we can pick some natural number n∉{n1,...,nk,m1,...ml}n∉{n1,...,nk,m1,...ml} and it is in both UU and VV.