Like many mathematical properties, paracompactness this is a notion of smallness. It's not about the smallness of a subset U of X, but smallness of a collection C of subsets U of X.
The definiton for a subset C of T to be locally finite says that you may consider a well choosen sample of neighborhoods (sets V∈T) and C ought to be finite with respect to that sample (i.e. finite pro V). Pic related for a concise definition.
A topologal space is paracompact if it has a cover with that local finiteness property.
The sample of V's above may be very big, so C is really only small w.r.t. the sample. In a compact space, on the other hand, the cover itself is finite (and you don't need to consider that sample).
Note that the name locally compact is already used for the situation where every point x∈X has a compact neighborhood V.
Question: What can we say, in general, about the notions of paracompact vs. locally compact ??
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?
John Adams
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?
Brandon Diaz
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.
David Richardson
Also, I think you messed the definition, paracompact is that *any* open cover has a locally finite refinement.
Jacob Jackson
This is right. A second-countable Hausdorff space is paracompact if it's locally compact, too.
This too!
Does anyone know what it takes for a paracompact space to be locally compact? I tried googling it, but got nothing.
Joshua Roberts
>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.
Juan Brooks
>In general locally compact 'looks' weaker I meant the opposite, it 'looks' stronger.
Adrian Carter
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.
