Locally finite collection problem

[radou]

Homework Statement

I'm not especially good at creating examples, so I'd like to check this one.

One needs to find a point-finite open covering of R which is not locally finite. (A collection is point-finite if each point of R lies in only finitely many elements of that collection)

The Attempt at a Solution

Let's define the collection A:

A = {<-n, 3-n> U <n, n + 2> U <0, 1/n> : n is in N}

Now, let's check A is point-finite. Let x be in R. Wherever x is, there exists some n such that x is not in the upper union of intervals indexed by n. (We can think of this as the interval <-n, 3-n> "escaping" to -∞, the interval <n, n + 2> "escaping" to +∞, and <0, 1/n> "escaping to 0, btw excuse me for these primitive notions).

Now, to see A is not locally finite, let U be any neighborhood of 0. Then it intersects the collection A in infinitely many elements, because of the interval <0, 1/n> contained in every element of A.

Thanks in advance for any comments.

 

[micromass]

This is entirely correct. It is actually a very nice example! Good job!

 

[radou]

OK, thanks!

 

[radou]

Btw, the next exercise seems to be a bit more sophisticated; to find a collection of sets A which is not locally finite, but so that the collection consisting of the closures of these sets is locally finite! Any hints on that one?