Question

Difference between interior of closure and closure of interior

Answer 1

A Comparison of the Interior and Closure of a Set

The Interior of A, int(A)The Closure of A, A¯The interior of A is the LARGEST OPEN set CONTAINED in AThe closure of A is the SMALLEST CLOSED set CONTAINING Aint(A)⊆AA⊆A¯A is OPEN if and only if A=int(A)A is CLOSED if and only if A=A¯If A⊆B then int(A)⊆int(B)If A⊆B then A¯⊆B¯int(A)∪int(B)⊆int(A∪B)A¯∪B¯=A∪B¯¯¯¯¯¯¯¯¯int(A)∩int(B)=int(A∩B)A¯∩B¯⊇A∩B¯¯¯¯¯¯¯¯¯

Answer 2

Answer:

Its interior is the set of all points that satisfy x2 + y2 + z2 < 1, while its closure is x2 + y2 + z2 <= 1. Therefore, the closure is the union of the interior and the boundary (its surface x2 + y2 + z2 = 1).