Moore-Smith Sequences

I skimmed through a newly acquired mathematics book, as I do, when I came across a section describing something I hadn’t seen before. As the title of this essay suggests, the section defined a Moore-Smith sequence, also known as a “net”, as a generalization of an index set for the usual sequences defined in undergraduate analysis.

I had heard of filters, or more broadly, ultra-filters, in writings I never understood, so I was intrigued to see a seemingly related concept show up in a familiar context.

As a quick recap, a sequence is an indexed set of numbers. For example, the numbers x_i: 1, 2, 3, 4, … form a sequence of increasing integers, where x_0 is 1, x_1 is 2, etc. The indices for the sequence x_1 are positive integers.

The Moore-Smith sequence answers the question, can you have a sequence whose index set is something other than positive integers? And the answer is yes. Here’s how Gerresten and Rau (translated by Gould) present the concept:

Let the index set be a (non-empty) set X of real numbers with a limit point a. The set becomes a directed set if we agree that p ~ q means |p - a| >= |q - a|. The [generalized metric] conditions are then easily verified. A mapping f of the set X into another set Y of real numbers, or in other words a real function, determines a Moore-Smith sequence.

Now, remember, if the index set were integers, we'd clearly have p < q unless p = q, giving a natural ordering on the positive integers. The generalization with the relation symbol ~ merely states the same concept. The key distinction is that the verification requires the concept of a limit point as a prerequisite. This seems pretty circular to me, as I'd normally expect the limit to be defined in terms of sequences!

But of course, it seems like here we don't redefine sequences so much as generalize the idea of sequences, and of metric spaces. Here's the generalized metric conditions referred by the quotation:

1. Transitive: If p~q and q~r then p~r

2. Complete: For any p and q in N, there's some r in N such that p~r and q~r

3. Identity: p~p for every p

In the case of the natural numbers, 1 and 2, you cannot find any number that's less than both; in the case of positive integers 0 and 1, the same situation holds. But provided reals 0 and 1, you can find a sequences of reals larger than zero that can be ordered by the conditions defined on ~.

This leads directly to a justification for epsilon-delta formulation of continuity:

In the sense of Moore-Smith convergence [the sequence of values of the function f(x)] has a limit b if for every assigned number epsilon there exits a number x_0 in X such that |f(x) - b| is less than epsilon for all x with x_0 ~ x, which means that |x_0 - a| >= |x - a|, or in other words |x - a|1 is less than or equal to delta with delta |x_0 - a|.

Think about it for a minute. If your sequence concept relies only on positive integers, you might have a difficult time thinking about how real inputs x to a function f(x) can create a sequence. There's seemingly more inputs than "positions" indexed by the positive integers. Normally, we'd need a clunky formulation of sequence values using n >= N where N is some value coming from the index set. But working out the generalization with the relation ~ gives us a way to write out the continuity proof without providing such an N.

Moreover, it seems (though I still do not understand how) that the Moore-Smith sequence allows for a set-theoretic construction, i.e. filters. That seems very cool and powerful, and very much in vogue for early 20th century approaches to mathematics.