Answer to Question #144074 in Differential Geometry | Topology for Ebrahim Sabah

Question

Q/Let X {1,2,3, 4} and T = {0,X,{1}, {1,2). {1,2,3), {1,3}, {1,3, 4}, where
(X,T) is topological space. If A = {1,3,4}, B = {2,3} then find A, B°, ext (A), ext(B), b (A), b(B)

Answer 1

$B^o$ = union of subsets of B open in X. But they are only $\emptyset.$ $A^o=$ $\{1,3,4\}$ since A is open.

ext(B)= union of open sets disjoint from B = $\{1\}.$ ext(A)=$\emptyset.$

b(B)= $\overline{B} \setminus B^o=\overline{B}.$ Now $\{1\}^c=\{2,3,4\}$. So the latter is closed and contains B. Also B is not closed since $B^{c}=\{1,4\}$ is not open. Hence $\{2,3,4\}$ is the smallest closed set containing B. Hence $b(B)=\overline{B}=\{2,3,4\}.$ $\overline{A}=X$ since only other option is $A$ itself and $A^c=\{2\}$ not open, hence A not closed. Hence b(A)=X$\setminus A^o=X\setminus A=\{2\}$.