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Problem. If $(x_i)$ is a net converging to a point $x$, show that $(x_i)$ converges to $x$ along any nonprincipal ultrafilter.
I will define these things below.
A directed set is a poset $(I,\leq)$ such that any two elements have an upper bound.
A net in a topological space $X$ is a func...
In the mathematical field of set theory, an ultrafilter on a given partially ordered set (poset) P is a certain subset of P, namely a maximal filter on P, that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P.
If X is an arbitrary set, its power set ℘(X), ordered by set inclusion, is always a Boolean algebra and hence a poset, and (ultra)filters on ℘(X) are usually called "(ultra)filters on X". An ultrafilter on a set X may be considered as a finitely additive measure on X. In this view, every subset of X is either considered "almost everything" (has measure 1)...
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I've noticed that one question was recently tagged filter. (It is the only question having this tag at the moment.)
I was thinking about a few related tags, which might perhaps be useful; but I wanted to ask about the opinion of other users before creating any of these tags. (I might be biased, ...
