So, find one open cover that cannot be reduced to a finite subcover?
But wouldn't that imply that I have to state what this open cover looks like? Should I maybe let R be an open cover of itself?
But the question states that I should do this proof directly from the definition of being compact. Ie, if an open cover can be reduced to a finite subcover.
A trivial, yet difficult question. How would one prove that the real numbers are not compact, only using the definition of being compact? In other words, what happens if we reduce an open cover of R to a finite cover of R?
I let V be a collection of open subset that cover R
Then I make the...