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: