<s>
In	O
mathematics	O
,	O
more	O
specifically	O
in	O
general	O
topology	B-Architecture
and	O
related	O
branches	O
,	O
a	O
net	O
or	O
Moore	O
–	O
Smith	O
sequence	O
is	O
a	O
generalization	O
of	O
the	O
notion	O
of	O
a	O
sequence	O
.	O
</s>
<s>
In	O
essence	O
,	O
a	O
sequence	O
is	O
a	O
function	O
whose	O
domain	B-Algorithm
is	O
the	O
natural	O
numbers	O
.	O
</s>
<s>
The	O
codomain	B-Algorithm
of	O
this	O
function	O
is	O
usually	O
some	O
topological	O
space	O
.	O
</s>
<s>
The	O
motivation	O
for	O
generalizing	O
the	O
notion	O
of	O
a	O
sequence	O
is	O
that	O
,	O
in	O
the	O
context	O
of	O
topology	B-Architecture
,	O
sequences	O
do	O
not	O
fully	O
encode	O
all	O
information	O
about	O
functions	O
between	O
topological	O
spaces	O
.	O
</s>
<s>
Nets	O
are	O
one	O
of	O
the	O
many	O
tools	O
used	O
in	O
topology	B-Architecture
to	O
generalize	O
certain	O
concepts	O
that	O
may	O
not	O
be	O
general	O
enough	O
in	O
the	O
context	O
of	O
metric	O
spaces	O
.	O
</s>
<s>
Any	O
function	O
whose	O
domain	B-Algorithm
is	O
a	O
directed	O
set	O
is	O
called	O
a	O
.	O
</s>
<s>
Elements	O
of	O
a	O
net	O
's	O
domain	B-Algorithm
are	O
called	O
its	O
.	O
</s>
<s>
Indeed	O
,	O
a	O
net	O
whose	O
domain	B-Algorithm
is	O
the	O
natural	O
numbers	O
is	O
a	O
sequence	O
because	O
by	O
definition	O
,	O
a	O
sequence	O
in	O
is	O
just	O
a	O
function	O
from	O
into	O
It	O
is	O
in	O
this	O
way	O
that	O
nets	O
are	O
generalizations	O
of	O
sequences	O
.	O
</s>
<s>
A	O
net	O
in	O
may	O
also	O
be	O
written	O
as	O
which	O
expresses	O
the	O
fact	O
that	O
this	O
net	O
is	O
a	O
function	O
whose	O
value	O
at	O
an	O
element	O
in	O
its	O
domain	B-Algorithm
is	O
denoted	O
by	O
instead	O
of	O
the	O
usual	O
parentheses	O
notation	O
that	O
is	O
typically	O
used	O
with	O
functions	O
(	O
this	O
subscript	O
notation	O
being	O
taken	O
from	O
sequences	O
)	O
.	O
</s>
<s>
As	O
in	O
the	O
field	O
of	O
algebraic	O
topology	B-Architecture
,	O
the	O
filled	O
disk	O
or	O
"	O
bullet	O
"	O
denotes	O
the	O
location	O
where	O
arguments	O
to	O
the	O
net	O
(	O
that	O
is	O
,	O
elements	O
of	O
the	O
net	O
's	O
domain	B-Algorithm
)	O
are	O
placed	O
;	O
it	O
helps	O
emphasize	O
that	O
the	O
net	O
is	O
a	O
function	O
and	O
also	O
reduces	O
the	O
number	O
of	O
indices	O
and	O
other	O
symbols	O
that	O
must	O
be	O
written	O
when	O
referring	O
to	O
it	O
later	O
.	O
</s>
<s>
Nets	O
are	O
primarily	O
used	O
in	O
the	O
fields	O
of	O
Analysis	O
and	O
Topology	B-Architecture
,	O
where	O
they	O
are	O
used	O
to	O
characterize	O
many	O
important	O
topological	O
properties	O
that	O
(	O
in	O
general	O
)	O
,	O
sequences	O
are	O
unable	O
to	O
characterize	O
(	O
this	O
shortcoming	O
of	O
sequences	O
motivated	O
the	O
study	O
of	O
sequential	O
spaces	O
and	O
Fréchet	O
–	O
Urysohn	O
spaces	O
)	O
.	O
</s>
<s>
Nets	O
are	O
intimately	O
related	O
to	O
filters	O
,	O
which	O
are	O
also	O
often	O
used	O
in	O
topology	B-Architecture
.	O
</s>
<s>
Every	O
net	O
may	O
be	O
associated	O
with	O
a	O
filter	O
and	O
every	O
filter	O
may	O
be	O
associated	O
with	O
a	O
net	O
,	O
where	O
the	O
properties	O
of	O
these	O
associated	O
objects	O
are	O
closely	O
tied	O
together	O
(	O
see	O
the	O
article	O
about	O
Filters	O
in	O
topology	B-Architecture
for	O
more	O
details	O
)	O
.	O
</s>
<s>
However	O
,	O
filters	O
,	O
and	O
especially	O
ultrafilters	O
,	O
have	O
some	O
important	O
technical	O
advantages	O
over	O
nets	O
that	O
ultimately	O
result	O
in	O
nets	O
being	O
encountered	O
much	O
less	O
often	O
than	O
filters	O
outside	O
of	O
the	O
fields	O
of	O
Analysis	O
and	O
Topology	B-Architecture
.	O
</s>
<s>
The	O
map	O
is	O
an	O
order	O
morphism	O
whose	O
image	O
is	O
cofinal	O
in	O
its	O
codomain	B-Algorithm
and	O
holds	O
for	O
every	O
This	O
shows	O
that	O
is	O
a	O
subnet	O
of	O
the	O
sequence	O
(	O
where	O
this	O
subnet	O
is	O
not	O
a	O
subsequence	O
of	O
because	O
it	O
is	O
not	O
even	O
a	O
sequence	O
since	O
its	O
domain	B-Algorithm
is	O
an	O
uncountable	O
set	O
)	O
.	O
</s>
<s>
Suppose	O
is	O
a	O
metric	O
space	O
(	O
or	O
a	O
pseudometric	O
space	O
)	O
and	O
is	O
endowed	O
with	O
the	O
metric	O
topology	B-Architecture
.	O
</s>
<s>
It	O
is	O
sometimes	O
useful	O
to	O
think	O
of	O
this	O
definition	O
in	O
terms	O
of	O
function	B-Application
composition	I-Application
:	O
the	O
net	O
is	O
equal	O
to	O
the	O
composition	O
of	O
the	O
net	O
with	O
the	O
projection	O
that	O
is	O
,	O
</s>
<s>
If	O
is	O
infinite	O
and	O
is	O
not	O
empty	O
,	O
then	O
the	O
axiom	O
of	O
choice	O
would	O
(	O
in	O
general	O
)	O
still	O
be	O
needed	O
to	O
conclude	O
that	O
the	O
projections	O
are	O
surjective	B-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
Every	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
and	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
can	O
be	O
interpreted	O
as	O
a	O
limit	O
of	O
a	O
net	O
(	O
as	O
described	O
below	O
)	O
.	O
</s>
<s>
Interpret	O
the	O
set	O
of	O
all	O
functions	O
with	O
prototype	O
as	O
the	O
Cartesian	O
product	O
(	O
by	O
identifying	O
a	O
function	O
with	O
the	O
tuple	O
and	O
conversely	O
)	O
and	O
endow	O
it	O
with	O
the	O
product	O
topology	B-Architecture
.	O
</s>
<s>
This	O
(	O
product	O
)	O
topology	B-Architecture
on	O
is	O
identical	O
to	O
the	O
topology	B-Architecture
of	O
pointwise	O
convergence	O
.	O
</s>
<s>
Virtually	O
all	O
concepts	O
of	O
topology	B-Architecture
can	O
be	O
rephrased	O
in	O
the	O
language	O
of	O
nets	O
and	O
limits	O
.	O
</s>
<s>
This	O
may	O
be	O
useful	O
to	O
guide	O
the	O
intuition	O
since	O
the	O
notion	O
of	O
limit	O
of	O
a	O
net	O
is	O
very	O
similar	O
to	O
that	O
of	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
It	O
is	O
these	O
characterizations	O
of	O
"	O
open	O
subset	O
"	O
that	O
allow	O
nets	O
to	O
characterize	O
topologies	B-Architecture
.	O
</s>
<s>
Topologies	B-Architecture
can	O
also	O
be	O
characterized	O
by	O
closed	O
subsets	O
since	O
a	O
set	O
is	O
open	O
if	O
and	O
only	O
if	O
its	O
complement	O
is	O
closed	O
.	O
</s>
<s>
So	O
the	O
characterizations	O
of	O
"	O
closed	O
set	O
"	O
in	O
terms	O
of	O
nets	O
can	O
also	O
be	O
used	O
to	O
characterize	O
topologies	B-Architecture
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
called	O
if	O
every	O
Cauchy	O
net	O
converges	O
to	O
some	O
point	O
.	O
</s>
<s>
A	O
normed	O
space	O
,	O
which	O
is	O
a	O
special	O
type	O
of	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
,	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
(	O
equivalently	O
,	O
a	O
Banach	O
space	O
)	O
if	O
and	O
only	O
if	O
every	O
Cauchy	O
sequence	O
converges	O
to	O
some	O
point	O
(	O
a	O
property	O
that	O
is	O
called	O
)	O
.	O
</s>
<s>
Although	O
Cauchy	O
nets	O
are	O
not	O
needed	O
to	O
describe	O
completeness	O
of	O
normed	O
spaces	O
,	O
they	O
are	O
needed	O
to	O
describe	O
completeness	O
of	O
more	O
general	O
(	O
possibly	O
non-normable	O
)	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
A	O
filter	O
is	O
another	O
idea	O
in	O
topology	B-Architecture
that	O
allows	O
for	O
a	O
general	O
definition	O
for	O
convergence	O
in	O
general	O
topological	O
spaces	O
.	O
</s>
<s>
For	O
instance	O
,	O
continuity	O
of	O
a	O
function	O
from	O
one	O
topological	O
space	O
to	O
the	O
other	O
can	O
be	O
characterized	O
either	O
by	O
the	O
convergence	O
of	O
a	O
net	O
in	O
the	O
domain	B-Algorithm
implying	O
the	O
convergence	O
of	O
the	O
corresponding	O
net	O
in	O
the	O
codomain	B-Algorithm
,	O
or	O
by	O
the	O
same	O
statement	O
with	O
filter	O
bases	O
.	O
</s>
<s>
He	O
argues	O
that	O
nets	O
are	O
enough	O
like	O
sequences	O
to	O
make	O
natural	O
proofs	O
and	O
definitions	O
in	O
analogy	O
to	O
sequences	O
,	O
especially	O
ones	O
using	O
sequential	O
elements	O
,	O
such	O
as	O
is	O
common	O
in	O
analysis	O
,	O
while	O
filters	O
are	O
most	O
useful	O
in	O
algebraic	O
topology	B-Architecture
.	O
</s>
<s>
In	O
any	O
case	O
,	O
he	O
shows	O
how	O
the	O
two	O
can	O
be	O
used	O
in	O
combination	O
to	O
prove	O
various	O
theorems	O
in	O
general	O
topology	B-Architecture
.	O
</s>
<s>
Some	O
authors	O
work	O
even	O
with	O
more	O
general	O
structures	O
than	O
the	O
real	O
line	O
,	O
like	O
complete	B-Algorithm
lattices	O
.	O
</s>
