Finite-Compliment Topology and intersection of interior

[math707]

[SOLVED]Finite-Compliment Topology and intersection of interior

Homework Statement

Given topological space (R^{1}, finite compliment topology), find counter example to show that

Arbitary Intersection of (interior of subset of R^{1}) is not equal to Interior of (arbitary intersection of subset of R^{1}).

\bigcap^{\infty}_{n=1}int(A_{n})\neq int(\bigcap^{\infty}_{n=1}A_{n})

Homework Equations

When we consider topological space (R^{1}, usual topology), it is easy to find out that [-1/n, 1/n] is the example.

The Attempt at a Solution

First, I thought what shapes the open of topological space (R^{1}, finite compliment topology) might have, and it seems to have some shape of real line having finite omissions.
And I tried A_{n} = R-[-1/n, 1/n], and [-1/n, 1/n], but found out all of these are not counterexample..

 

[morphism]

Look at random subsets of R, and study what their interiors look like.

 

[math707]

A(n) = R-{n}

A(n)=R-{n} is the counterexample...
I tried this case many times but it was not the counterexample,
but after some fresh air, it is clear this is counter example..