Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Can one give an example of non-compact space $X$ which satisfies the following conditions:

  • the countable union of compact subsets is relatively compact,

  • for every closed noncompact subset $A$ of $X$ there is a positive lower semicontinuous function on $X$ which is bounded on every compact subset of $X$ but unbounded on $A$.

Thanks in advance for any help.

share|improve this question
    
en.wikipedia.org/wiki/Counterexamples_in_Topology would be a good reference. –  David Roberts Sep 6 '11 at 0:17
1  
Any reason why you're interested in such a space? That may help to shape answers. –  David Roberts Sep 6 '11 at 0:18
    
If $X$ is like this then $C(X)$ with compact-open topology is a quasi-barrelled (DF)-space. I'm looking for the example of the lcs space which is like this, but is not normed. –  Celeban Sep 6 '11 at 1:36
add comment

1 Answer

up vote 4 down vote accepted

I believe such a space cannot exist for the following reason:

Suppose it does. By the second requirement, there should be an unbounded function $f: X \to (0, \infty)$, which is bounded on every compact set. This means we have countably many points $x_1, x_2, x_3, \dots$ such that for each $n$ the inequality $f(x_n)\ge n$ holds. The singletons $\lbrace x_n\rbrace$ are finite sets, therefore compact. By the first requirement, the set $\lbrace x_n| n\in\mathbb{N}\rbrace$ is therefore relatively compact and so its closure must be compact. But then $f$ is unbounded on a compact set, which is a contradiction.

share|improve this answer
    
Thanks, it seems to be correct. –  Celeban Sep 6 '11 at 14:08
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.