- At the beginning of proof of problem 2, The author choose arbitrary a in (0,1) and
Claimed that there exists some closed interval [a1,a2] containing a.
I know this is right, but I'm curious that this is always guranteed or is just specific case.
Is it possible for some open sets not to have a compact subset (except singleton) ?
- The author showed that f is continuous on [a1,a2] and this set is arbitrary.
But does it gurantee that if f is continuous on all of such compact sets, then it follows that f is continuous on open set such as (0,1)? Doesn't exist some point of (0,1) not in such compact sets?
I think this questions can be summarized as can open set be expressed as an union of compact sets?
If I missed something, could you point it out?
It will be very helpful to solidate my understading about elementary topology.
Thank you for your answer in advance.