View source for Net (directed set)

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A mapping of a [[Directed set|directed set]] into a (topological) space.

====Comments====
The topology of a space can be described completely in terms of convergence. However, this needs a more general concept of convergence than the concept of convergence of a sequence. What is needed is convergence of nets. A net  $  S :  D \rightarrow X $
in a topological space  $  X $
converges to a point  $  s \in X $
if for each open neighbourhood  $  U $
of  $  s $
in  $  X $
the net  $  S $
is eventually in  $  U $. 
The last phrase means that there is an  $  m \in D $
such that  $  S ( n) \in U $
for all  $  n \geq  m $
in  $  D $.

The theory of convergence of nets is known as Moore–Smith convergence, [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , v. Nostrand  (1955)  pp. Chapt. II</TD></TR></table>