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partition of unity, locally finite cover
Will put up some stuff about Dold’s trick of taking a not-necessarily point finite partition of unity and making a partition of unity. There is a case when I know it works and a case I’m really not sure about - I need to find where the argument falls down because I get too strong a result. I’ll discuss this in the thread soon, and port it over when it is stable.
I’ve found the argument that I was discussing with Harry here, in Dold’s lectures on algebraic topology. It’s a result due to an M. Mather, in a 1965 Cambridge PhD thesis, Paracompactness and partitions of unity. Will type it here then copy it over to the lab when at work tomorrow.
Definition: A collection of functions $u_i : X \to [0,1]$ is called locally finite if the cover $u_i^{-1}(0,1]$ (the induced cover) is locally finite.
Proposition (Mather, 1965): Let $\{u_i\}_J$ be a non-point finite partition of unity. Then there is a locally finite partition of unity $\{v_i\}_{i\in J}$ such that the induced cover of the latter is a refinement of the induced cover of the former.
’Dold’s trick’ is about taking a countable family of functions $u_i$ and turning it into a locally finite partition of unity, the proof of the above proposition is a little bit different in flavour (but not that different).
copy it over to the lab
I’ve now put this in at partition of unity.
What’s a “non-point finite partition of unity”? If it is what it sounds like, then how is the sum well defined?
I have added to locally finite cover statement and proof that every locally finite refinement induces a locally finite cover with the original index set.
Hmm, not sure. I’d have to check Dold’s book again. In principle one could say that one has a collection of functions to [0,1] such that at each point only countably many are nonzero, and the sum exists and is 1 at each point.
Yes, it seems to be as I thought, from looking at the Google Books link, just before Lemma 2.6 in the Appendix.
Okay, thanks. We should says this more clearly in the entry then.
If anyone is interested, there is a constructive (BISH) treatment of ’partition of unity’, ’locally finite open cover’, and ’star-finite open cover’ in section 3.1 of my thesis modern intuitionistic topology. It gives a simple proof that every per-enumerable* cover of a separable metric space has i) a star-finite refinement b) a subordinate partition of unity.
*A subset $U$ is enumerably open when it is an enumerable union of basic opens, a cover $\mathcal{U}$ is per-enumerable when it is an enumerable collection of enumerably open sets.
Oh well, that’s nice!, thank you. I might edit the entry in ‘partition of unity’ since it now reads ’in intuitionistic mathematics’ whereas the relevant definitions and proofs are already in BISH (constructive mathematics, without use of intuitionistic axioms).
{Many paragraphs in the thesis are marked with an asterisk * to indicate that they are valid in BISH}.
I edited partition of unity to clarify around the case without point-finiteness.
David, thanks! Where you wrote “convergent infinite sum” I made it come out as “convergent infinite series”.
Frank, okay, thanks.
Presently the $n$Lab pages intuitionstic mathematics and constructive mathematics may not completely reflect the sharp distinction that you are making here (or if they do, I find it hard to extract it). Maybe you have the energy to touch these pages accordingly?
OK, fair enough, I’ll see what I can do in the coming month. I’m always a bit reluctant to edit because I’m not sure that my ‘style’ will fit in or serve the purpose of the nLab. But I should be able to manage some clarification and specification of ideas and axioms. It’s true that my energy is limited, so it will take a bit more time than average, but I suppose that won’t be a problem. [By the way, I find it really really impressive how hard and conscientiously people work on nLab, and work together.]
I’ll notify of any changes I make, here on nForum.
Thanks, Frank! Please don’t worry about style. The important point is the content.
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