Open cover which has no finite subcover

[tracedinair]

Homework Statement

Give an example of an open cover in R^n which has no finite subcover.

Homework Equations

The Attempt at a Solution

{x ε Q | x < sqrt(3)} U {x ε Q | x > sqrt(3) }

 

[Hurkyl]

None of the elements in that set is an open subset of Rⁿ. None of them are even subsets of Rⁿ!


Or... were you trying to say that you wanted to consider an open cover whose elements are the two sets

{x ε Q | x < sqrt(3)} and {x ε Q | x > sqrt(3) }​

? If so, you need to show four things:

(1) Those two sets are subsets of Rⁿ
(2) Those two sets are open
(3) Those two sets cover Rⁿ
(4) No finite subcover covers the same space as the entire cover does

 

[tracedinair]

n/m. I've figured this out.