ultranet

A net ${(x_a)}_{a\in A}$ on a set $X$ is said to be an ultranet or universal net if whenever $E\subseteq X$, $(x_a)$ is either eventually in $E$ or eventually in $X\setminus E$.

:

• It can be shown that every net has a universal subnet.

• When $X$ is a locally compact topological space, a universal net in $X$ is either convergent or it “goes to ” (it eventually leaves every compact subset).

Title	ultranet
Canonical name	Ultranet
Date of creation	2013-03-22 12:54:35
Last modified on	2013-03-22 12:54:35
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	9
Author	asteroid (17536)
Entry type	Definition
Classification	msc 54A20
Synonym	universal net
Related topic	Ultrafilter
Related topic	EveryNetHasAUniversalSubnet