I just want to know what fails when using the same argument of Heine-Borel for trying to prove that an open interval $(a, b)$ is compact.
For a given cover $\{U_i\}$, let $A = \{x \in (a, b) ;(a, x)$ covered by a finite number of $U_i \}$, then, using the same argument as in Heine-Borel (for $[a, b]$) and assuming that $supA> a$, I get that $supA = b$. So, is there anything wrong in this proof (idea)? Is the possibility of $supA = a$ the only reason for $(a, b)$ not being compact?
Thanks in advance.