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k-spaces can be characterized as the topological inductive limit of compact spaces. Locally compact spaces are k-spaces but not conversely (k-spaces can also be characterized as quotients of locally compact spaces).

Locally compact spaces have the distinguished property that they have a bunch of relatively compact open sets: every compact set is contained in a relatively compact open set and moreover that if $K$ is compact and $U$ is open with $K \subseteq U$ then one can find an open set $U'$ and a compact set $K'$ such that $K \subseteq U' \subseteq K' \subseteq U$. Do these properties completely characterize locally compact spaces?

A locally compact space is also the topological inductive limit of its relatively compact open sets. Can this be used to characterize locally compact spaces as an inductive limit of particular systems of topological spaces?

yada
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  • Do you assume Hausdorff here? If not, then what definition of local compactness you use? – freakish Nov 28 '19 at 10:30
  • Yes, locally compact spaces are assumed to be Hausdorff here. – yada Nov 28 '19 at 10:31
  • Do I understand correctly that you want to specify a certain class of direct systems of compact spaces such that each locally compact space is the direct limit of such a system? – Paul Frost Nov 28 '19 at 18:40
  • @Paul Frost Exactly. Also, any locally compact space is the limit of its relatively compact open sets. So this seems to impose restrictions on the directed system. – yada Nov 28 '19 at 19:27
  • It's quite easy to see that within Hausdorff spaces your $K,U,K',U'$ property is equivalent to local compactness. – Henno Brandsma Nov 28 '19 at 22:14
  • @HennoBrandsma Yes, indeed. The focus was more on the inductive limit characterization. I mentioned the $K, U, K', U'$ property to show up the potential restrictions on the inductive systems. – yada Nov 29 '19 at 07:02

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