Homework Question: Understanding a Lemma on Open Coverings in Regular Spaces

[radou]

Homework Statement

OK, just a short question about a lemma I'm going through in Munkres and a part of its proof.

I won't quote the whole lemma (it's a few statements which are equivalent), but only the part I don't get:

Let X be regular. If every open covering of X has a refinement that is an open covering of X and locally finite, then every open covering of X has a refinement that is an open covering of X and countably locally finite.

In the proof it states that this is trivial, so I'm missing something obvious apparently. But I just can't see what.

Thanks in advance...

 

[micromass]

I think you're thinking to far here. The proof is really simple: take an covering $$\mathcal{B}$$ which is open and locally finite. Then the covering is also countable locally finite. Thus we have found an open cover which is countable locally finite...

 

[radou]

I think you're thinking to far here. The proof is really simple: take an covering $$\mathcal{B}$$ which is open and locally finite. Then the covering is also countable locally finite. Thus we have found an open cover which is countable locally finite...

Ahhhh this is utterly trivial ! Since this cover (call it A) can be written as a union A = U A (this doesn't make sense, but I think it's clear what I'm trying to say) consisting of a single element, that cover itself, which is locally finite! :uhh:

Sorry, sometimes I get confused on a really stupid base.

 

[micromass]

Haha, looks like somebody partied a little too hard last night

 

[radou]

Haha, looks like somebody partied a little too hard last night

Yes, one could say...

Btw, the proof of the implication (3) ==> (4) in the same lemma (i.e. if every open covering of X has a refinement that is a closed locally finite covering of X, then every open covering of X has a refinement that is an open locally finite covering of X) is beautiful, at least to me. At a first glance, it seems a bit complicated, but when you actually think about what's going on, it's great!

 

[micromass]

Yeah, it is quite beautiful. It reminds me a bit of measure theory: you do some complicated things for complicated reasons, and it all ends really nice

When I first learned about it, I found paracompactness quite complicated. But it has it's beauty...