<s>
In	O
mathematics	O
,	O
pointwise	B-Algorithm
convergence	I-Algorithm
is	O
one	O
of	O
various	B-Algorithm
senses	I-Algorithm
in	O
which	O
a	O
sequence	O
of	O
functions	O
can	O
converge	B-Algorithm
to	O
a	O
particular	O
function	O
.	O
</s>
<s>
It	O
is	O
weaker	O
than	O
uniform	B-Algorithm
convergence	I-Algorithm
,	O
to	O
which	O
it	O
is	O
often	O
compared	O
.	O
</s>
<s>
Sometimes	O
,	O
authors	O
use	O
the	O
term	O
bounded	O
pointwise	B-Algorithm
convergence	I-Algorithm
when	O
there	O
is	O
a	O
constant	O
such	O
that	O
.	O
</s>
<s>
This	O
concept	O
is	O
often	O
contrasted	O
with	O
uniform	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
That	O
is	O
a	O
stronger	O
statement	O
than	O
the	O
assertion	O
of	O
pointwise	B-Algorithm
convergence	I-Algorithm
:	O
every	O
uniformly	B-Algorithm
convergent	I-Algorithm
sequence	O
is	O
pointwise	O
convergent	O
,	O
to	O
the	O
same	O
limiting	O
function	O
,	O
but	O
some	O
pointwise	B-Algorithm
convergent	I-Algorithm
sequences	I-Algorithm
are	O
not	O
uniformly	B-Algorithm
convergent	I-Algorithm
.	O
</s>
<s>
The	O
pointwise	O
limit	O
of	O
a	O
sequence	O
of	O
continuous	O
functions	O
may	O
be	O
a	O
discontinuous	O
function	O
,	O
but	O
only	O
if	O
the	O
convergence	B-Algorithm
is	O
not	O
uniform	O
.	O
</s>
<s>
The	O
values	O
of	O
the	O
functions	O
need	O
not	O
be	O
real	O
numbers	O
,	O
but	O
may	O
be	O
in	O
any	O
topological	O
space	O
,	O
in	O
order	O
that	O
the	O
concept	O
of	O
pointwise	B-Algorithm
convergence	I-Algorithm
make	O
sense	O
.	O
</s>
<s>
Uniform	B-Algorithm
convergence	I-Algorithm
,	O
on	O
the	O
other	O
hand	O
,	O
does	O
not	O
make	O
sense	O
for	O
functions	O
taking	O
values	O
in	O
topological	O
spaces	O
generally	O
,	O
but	O
makes	O
sense	O
for	O
functions	O
taking	O
values	O
in	O
metric	O
spaces	O
,	O
and	O
,	O
more	O
generally	O
,	O
in	O
uniform	O
spaces	O
.	O
</s>
<s>
As	O
described	O
in	O
the	O
article	O
on	O
characterizations	O
of	O
the	O
category	O
of	O
topological	O
spaces	O
,	O
if	O
certain	O
conditions	O
are	O
met	O
then	O
it	O
is	O
possible	O
to	O
define	O
a	O
unique	O
topology	O
on	O
a	O
set	O
in	O
terms	O
of	O
what	O
nets	O
do	O
and	O
do	O
not	O
converge	B-Algorithm
.	O
</s>
<s>
A	O
net	O
in	O
converges	B-Algorithm
in	O
this	O
topology	O
if	O
and	O
only	O
if	O
it	O
converges	B-Algorithm
pointwise	O
.	O
</s>
<s>
The	O
topology	O
of	O
pointwise	B-Algorithm
convergence	I-Algorithm
is	O
the	O
same	O
as	O
convergence	B-Algorithm
in	O
the	O
product	O
topology	O
on	O
the	O
space	O
where	O
is	O
the	O
domain	O
and	O
is	O
the	O
codomain	B-Algorithm
.	O
</s>
<s>
If	O
the	O
codomain	B-Algorithm
is	O
compact	O
,	O
then	O
by	O
Tychonoff	O
's	O
theorem	O
,	O
the	O
space	O
is	O
also	O
compact	O
.	O
</s>
<s>
In	O
measure	O
theory	O
,	O
one	O
talks	O
about	O
almost	O
everywhere	O
convergence	B-Algorithm
of	O
a	O
sequence	O
of	O
measurable	O
functions	O
defined	O
on	O
a	O
measurable	O
space	O
.	O
</s>
<s>
That	O
means	O
pointwise	B-Algorithm
convergence	I-Algorithm
almost	O
everywhere	O
,	O
that	O
is	O
,	O
on	O
a	O
subset	O
of	O
the	O
domain	O
whose	O
complement	O
has	O
measure	O
zero	O
.	O
</s>
<s>
Egorov	O
's	O
theorem	O
states	O
that	O
pointwise	B-Algorithm
convergence	I-Algorithm
almost	O
everywhere	O
on	O
a	O
set	O
of	O
finite	O
measure	O
implies	O
uniform	B-Algorithm
convergence	I-Algorithm
on	O
a	O
slightly	O
smaller	O
set	O
.	O
</s>
<s>
Almost	O
everywhere	O
pointwise	B-Algorithm
convergence	I-Algorithm
on	O
the	O
space	O
of	O
functions	O
on	O
a	O
measure	O
space	O
does	O
not	O
define	O
the	O
structure	O
of	O
a	O
topology	O
on	O
the	O
space	O
of	O
measurable	O
functions	O
on	O
a	O
measure	O
space	O
(	O
although	O
it	O
is	O
a	O
convergence	B-Algorithm
structure	O
)	O
.	O
</s>
<s>
For	O
in	O
a	O
topological	O
space	O
,	O
when	O
every	O
subsequence	O
of	O
a	O
sequence	O
has	O
itself	O
a	O
subsequence	O
with	O
the	O
same	O
subsequential	O
limit	O
,	O
the	O
sequence	O
itself	O
must	O
converge	B-Algorithm
to	O
that	O
limit	O
.	O
</s>
<s>
Then	O
any	O
subsequence	O
of	O
the	O
sequence	O
has	O
a	O
sub-subsequence	O
which	O
itself	O
converges	B-Algorithm
almost	O
everywhere	O
to	O
zero	O
,	O
for	O
example	O
,	O
the	O
subsequence	O
of	O
functions	O
which	O
do	O
not	O
vanish	O
at	O
But	O
at	O
no	O
point	O
does	O
the	O
original	O
sequence	O
converge	B-Algorithm
pointwise	I-Algorithm
to	O
zero	O
.	O
</s>
<s>
Hence	O
,	O
unlike	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
and	O
convergence	B-Algorithm
,	O
pointwise	B-Algorithm
convergence	I-Algorithm
almost	O
everywhere	O
is	O
not	O
the	O
convergence	B-Algorithm
of	O
any	O
topology	O
on	O
the	O
space	O
of	O
functions	O
.	O
</s>
