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It seems that there are different conventions in the literature as to what is a locally compact space (when the space is not supposed Hausdorff).

The two main non equivalent definitions I've seen are :

  • (LC1) every point has a compact neighborhood

  • (LC2) every neighborhood of any point contains a compact neighborhood of the point.

I am wondering if there is a reason as to why one might prefer one definition to another. More precisely, in practice, what definition is really useful (yields interesting results), or are they are both important in their own right ? In that case why not give a different name to those definitions ?

Naively, when one looks at the definition of locally connected space, one does not use the version (LCn1)"every point has a connected neighborhood", but always (LCn2) "every neighborhood of any point contains a connected neighborhood of the point". Is there a deep reason as to why LCn1 is never considered, but LC1 is ?

I am aware that for Hausdorff space, LC1 and LC2 are equivalent, and since LC1 is easier to check, one might prefer that as a definition, but this argument is unconvincing if LC2 is actually more useful for non-Hausdorff space.

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    $\begingroup$ LCn1 seems clearly "wrong" since it is satisfied by every connected space, even those such as comb space or the topologist's sine curve which violate the whole idea of local connectedness. $\endgroup$
    – Nate Eldredge
    Commented Feb 25, 2017 at 7:26
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    $\begingroup$ You are probably aware that in most french books (following Bourbaki, and including Wikipedia), a locally compact space is Hausdorff by definition -- with good reasons in my view. $\endgroup$
    – abx
    Commented Feb 25, 2017 at 10:14
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    $\begingroup$ @abx : what are those good reasons ? $\endgroup$
    – Phil-W
    Commented Feb 25, 2017 at 10:41
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    $\begingroup$ The reason why LC2 and LCn2 are the "right" definitions is because they actually capture what "locally X" means. We don't care about "a" neighbourhood. Things hold locally if they hold on arbitrary small neighbourhoods. That LC1 is equivalent to LC2 in Hausdorff spaces is simply an "accident" that is rooted in the non-obvious interplay between compactness, separation axioms ($T_2+compact\implies T_3$) and a similarly accidental side effect of $T_3$ being equivalent to "locally closed". $\endgroup$
    – Johannes Hahn
    Commented Feb 25, 2017 at 11:03
  • $\begingroup$ @Phil-W: compare with the 3 possible definitions given by the english Wikipedia page, which concludes by "In almost all applications, locally compact spaces are indeed also Hausdorff". $\endgroup$
    – abx
    Commented Feb 25, 2017 at 11:11

2 Answers 2

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To me, the second definition of local compactness is much to be preferred for the simple reason that such locally compact spaces $X$ are exponentiable in $Top$, meaning that $X \times -: Top \to Top$ has a right adjoint $(-)^X: Top \to Top$ (even without the Hausdorff condition), and all this implies (such as $X \times -$ preserving coequalizers). In fact the necessary and sufficient condition for exponentiability, called core compactness, is only a mild generalization of local compactness (and equivalent to it under the Hausdorff assumption).

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    $\begingroup$ Similarly for local connectedness of $X$: one may argue that rather than being "a thing in itself" it is "needed" for the constant sheaf functor $\mathrm{Sets}\to\mathrm{Sheaves}(X)$ to have a left adjoint. And for that, LCn1 is useless, one really needs LCn2. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 25, 2017 at 9:46
  • $\begingroup$ @მამუკაჯიბლაძე Great observation! $\endgroup$
    – Todd Trimble
    Commented Feb 25, 2017 at 12:13
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The notion LCn1 just boils down to "the connectedness components of the space are clopen". If this property does indeed show up somewhere, I would expect that the latter is a more convenient way of expressing it.

LC1 on the other hand does indeed seem to capture some intuition about "this space has some properties of compact spaces, but might be too large to be compact". Naming it locally compact is probably a mistake coming from the equivalence to actual local compactness in Hausdorff spaces.

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  • $\begingroup$ Well with the latter intuition on the other hand it turns out counterintuitive that every compact space is LC1... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 25, 2017 at 15:42
  • $\begingroup$ I rephrased the statement to not exclude the compact spaces. $\endgroup$
    – Arno
    Commented Feb 25, 2017 at 15:50

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