What is the intersection of all the closed intervals contain

What is the intersection of all the closed intervals containing the open interval (0,1)? Justify your answer.

Solution

Every element of [0,1] is also an element of (0,1)

since the set of interior points of [0,1] is (0,1) and the set of boundary points consist of two end points

{0,1}. The accumulation points of [0,1] is [0,1].

Therefore (0,1) and the closed set of points [a,b] both have the same interior, isolated, boundary and accumulation points and also we observe that [a.b] contains boundary points but (0,1) doesnot.

Therefore any closed superset of [a,b] contains the open interval (0,1) and [0,1]

Therefore the intersection of all the closed intervals containing the open interval (0,1) is [0,1].