Question #144074
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)
1
Expert's answer
2020-11-13T15:57:09-0500

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

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

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


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