# Finite-Compliment Topology and intersection of interior

1. Oct 2, 2007

### math707

[SOLVED]Finite-Compliment Topology and intersection of interior

1. The problem statement, all variables and given/known data

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$$^{1}$$).

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

2. Relevant equations

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

3. 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..

Last edited: Oct 3, 2007
2. Oct 3, 2007

### morphism

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

3. Oct 3, 2007

### 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..