# 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 $X\smallsetminus E$.

:

• 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 Ultranet 2013-03-22 12:54:35 2013-03-22 12:54:35 asteroid (17536) asteroid (17536) 9 asteroid (17536) Definition msc 54A20 universal net Ultrafilter EveryNetHasAUniversalSubnet