PARACOMPACT vs. LOCALLY FINITE

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You talk about locally compact but your topic and the highlighted text in your pic talk about local finiteness. Which one is the property you are interested about?

>>8344705
Okay, I switched up the title.

I'm interested in
1. paracompact (where the definition involves local finiteness)
vs.
2. locally compact.

E.g. you you consdier the classes of paracompact spaces and the class of locally compact spaces, who do they relate to each other, which is the stronger requirement, what is to add to make on imply the other (my main interested in the quesiton)? And finally, in which cases are they the same?

>>8344682
Neither condition implies the other I think.
Metric implies paracompact. There are metric spaces which are not locally compact, example any infinite dimensional banach space.
A locally euclidean hausdorff space is metric iff it is paracompact iff the connected components are second countable. So a locally euclidean hausdorff connected space which is not normal (example in Bredon's topology) is not paracompact.
Locally compact hausdorff sigma-compact implies paracompact, though.

>>8344682
Also, I think you messed the definition, paracompact is that *any* open cover has a locally finite refinement.

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>>8345629
This is right. A second-countable Hausdorff space is paracompact if it's locally compact, too.

>>8345631
This too!

Does anyone know what it takes for a paracompact space to be locally compact? I tried googling it, but got nothing.

>>8345658
>Does anyone know what it takes for a paracompact space to be locally compact?
Don't think there is anything. In general locally compact 'looks' weaker, the counterexample is fairly delicate and convoluted, and a fairly tame countability condition rules it out. The counterexample for the other direction (e.g. banach spaces) is easier, and has good topological properties.

>>8345680
>In general locally compact 'looks' weaker
I meant the opposite, it 'looks' stronger.

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>>8345684
And this is why I was wondering if anyone knew some amplifier for paracompactness. I mean, a sequelly compact metrizable space is compact. The metrizability amplifies the notion of sequential compactness.

>>8345704
What do you mean by amplifier?

>>8344682
This: http://math.stanford.edu/~conrad/diffgeomPage/handouts/paracompact.pdf
shows that second countable, Hausdorff and locally compact implies paracompact.

>>8344682
>>8345707
Still me. I think the following provides a broad class of counterexamples to one direction (if it is correct): Take any topological space X having infinitely many open sets, and with the property that any two open sets intersect in a non-trivial way. Now add a point p to X and define a subset of this new set to be open if, and only if it is an open set of X with p added to it. The assumptions I have made above are in order to force the singleton {p} to not be open. Then Xu{p} is not paracompact.

If X was compact, Xu{p} is also compact, so if X was locally compact, then Xu{p} also is.

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>>8345705
Like in my example regarding sequentially compact spaces, how an additional assumption makes a strictly weaker property equal imply some other.

>>8345725
Seems to work without need for the additional point. Of course you'll have to prove something like that X exists, which is actually not that hard. But such a space is never hausdorff, though.

>>8345740
I don't follow. If metric is analogous to paracompact in your analogy, what is compactness and sequential compactness?

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>>8345750
Metric is actually the thing I'm after, sequential compactness is paracompactness and compactness is local compactness. What do we need to assume to have paracompact spaces locally compact? A quick thought process reduced it to another problem:
>assuming a space is Hausdorff and paracompact, what do we need to make closed sets compact?
Naturally, we could assume the space itself is compact, and this would give the result. The reasoning here is based on the fact that paracompact Hausdorff spaces are normal, which is equivalent to that, every point [math]x[/math] and every open nbd [math]U[/math], there is an open nbd [math]V[/math] of [math]x[/math] such that [math]V \subset \bar{V} \subset U[/math]. Since the space would be Hausdorff, the compactness of V's closure would make it locally compact.

>>8345770
Seems to be the same question as here then
>>8345658
I still think this
>>8345680
There shouldn't be a natural one.

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>>8345780
Yeah, there probably isn't. I just wouldn't say it so surely because of these cases like this metric and connectedness together with local connectedness implying path connectedness. I still lean to the direction that there is none.

>>8345745
Right. I was thinking of something like an algebraic variety over the complex numbers, of generic dimension at least 1, with the Zariski topology should do the job. But I think we do need the additional point, so that if we need an infinite number of opens in a cover to cover X, then any refinement will cover p an infinite number of times.

Are these fancy words just jargon to give unnecessarily tedious definitions to simple stuff?

>>8346603
If you're not a mathematician, then yes.
If you are a mathematician, sometimes you need some technical assumption to make things work. For example, usually the definition of (topological) manifold is a paracompact, Hausdorff, locally euclidean topological space.