Question

In topological space (N, T) where N is set of all natural numbers and
T {0, N, A, = {1,2,3, .., n}: n ∈ N}
Let A = {1, 2,4,6}, B = {5,7,9, 20}, find
A°, ext(A), b (A), B°, ext (B), b (B)

Answer 1

$A^o$ = union of open subsets of A. Hence $A^o= \{1,2\}\cup\{1\}\cup \{ \emptyset \}=\{1,2\}.$ ext(A)= union of open sets disjoint from A=$\emptyset$ since every nonempty set contains 1. b(A)= $\overline{A}\setminus A^o= \mathbb{N}\setminus \{1,2\}$. since ext(A) is empty, closure of A is the entire space.

$B^o=$ $\emptyset$ since all non-empty open sets contain 1. ext(B)=$=\emptyset\cup \{1\}\cup\{1,2\}\cup\{1,2,3\}\cup\{1,2,3,4\}$ $=\{1,2,3,4\}.$ b(B)= $\overline{B}\setminus B^o= \overline{B}.$ Since $\{1,2,3,4\}$

is the largest open set disjoint from B. Hence $\overline {B}=\mathbb{N} \setminus\{1,2,3,4\}.$