For a collection On of open sets where n 123 the intersect

For a collection {On} of open sets, where n = 1,2,3, ..., the intersection of On is open

Solution

A set A is open if every xA, there exists some >0 such that

B(x,){y:yC,d(x,y)<}A i.e. x is in some open ball that is in A, i.e. every xA is an interior point.

Let X=O1O2. Then any xX implies xO1 and xO2, implying that you should always have an open ball around x.

We prove via contradiction. Now assume X is not open. Since X is not open, there exists xXsuch that x is on the boundary of X. This implies that x is on the boundary of either O1 or O2. But this is a contradiction, since x has to be an interior point of both O1 and O2. Therefore, all elements of X are interior points. Therefore, X must be open.