Topological space with some conditions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:20:59Z http://mathoverflow.net/feeds/question/74604 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74604/topological-space-with-some-conditions Topological space with some conditions Celeban 2011-09-05T18:00:16Z 2011-09-06T01:57:01Z <p>Can one give an example of non-compact space $X$ which satisfies the following conditions:</p> <ul> <li><p>the countable union of compact subsets is relatively compact,</p></li> <li><p>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 $A$.</p></li> </ul> <p>Thanks in advance for any help.</p> http://mathoverflow.net/questions/74604/topological-space-with-some-conditions/74628#74628 Answer by Dejan Govc for Topological space with some conditions Dejan Govc 2011-09-06T01:40:17Z 2011-09-06T01:57:01Z <p>I believe such a space cannot exist for the following reason:</p> <p>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.</p>