Real Analysis Question on Fine-ness

[jacobsh47]

I'm having trouble understanding what makes a family fine. My book's definition says:
The family F is fine when, for each positive rational epsilon, there is an interval in F that has length less than epsilon.

We're also given the example that [1/n[sup:3pe35oh8]2[/sup:3pe35oh8] , 3/n] , n=1,2,... is fine because if we choose n so that n>3/epsilon the interval will have length less than epsilon.

I don't understand how to figure out whether or not an interval is fine. What would be the difference between a fine and a non-fine interval?

I'd really appreciate some help.

 

[daon]

I've never seen this definition, but I have seen finess used in partitioning.

It seems to me the defn is saying is that for all epsilon > 0, there is an interval S in F such that |S|<epsilon.

It is equivilant to saying there is always an interval of arbitrarily small length contained in F. A non-fine set is any finite collection of intervals... choose epsilon as the minimum of the lengths (over 2 if necessary). If F contains any singleton points, this is automatically true. Another example of a non-fine family would be F={(1/n,n+1), n natural #}. The smallest interval in this collection has length 1 (therefore epsilon=1/2 fails).