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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.

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  • $\begingroup$ en.wikipedia.org/wiki/Counterexamples_in_Topology would be a good reference. $\endgroup$
    – David Roberts ♦
    Commented Sep 6, 2011 at 0:17
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    $\begingroup$ Any reason why you're interested in such a space? That may help to shape answers. $\endgroup$
    – David Roberts ♦
    Commented Sep 6, 2011 at 0:18
  • $\begingroup$ 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. $\endgroup$
    – Celeban
    Commented Sep 6, 2011 at 1:36

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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.

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