<s>
In	O
mathematics	O
compact	B-Algorithm
convergence	I-Algorithm
(	O
or	O
uniform	B-Algorithm
convergence	I-Algorithm
on	I-Algorithm
compact	I-Algorithm
sets	I-Algorithm
)	O
is	O
a	O
type	O
of	O
convergence	B-Algorithm
that	O
generalizes	O
the	O
idea	O
of	O
uniform	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
It	O
is	O
associated	O
with	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
uniformly	B-Algorithm
on	O
as	O
.	O
</s>
<s>
If	O
and	O
with	O
their	O
usual	O
topologies	O
,	O
with	O
,	O
then	O
converges	O
compactly	O
to	O
the	O
constant	O
function	O
with	O
value	O
0	O
,	O
but	O
not	O
uniformly	B-Algorithm
.	O
</s>
<s>
If	O
,	O
and	O
,	O
then	O
converges	O
pointwise	B-Algorithm
to	O
the	O
function	O
that	O
is	O
zero	O
on	O
and	O
one	O
at	O
,	O
but	O
the	O
sequence	O
does	O
not	O
converge	O
compactly	O
.	O
</s>
<s>
A	O
very	O
powerful	O
tool	O
for	O
showing	O
compact	B-Algorithm
convergence	I-Algorithm
is	O
the	O
Arzelà	O
–	O
Ascoli	O
theorem	O
.	O
</s>
<s>
There	O
are	O
several	O
versions	O
of	O
this	O
theorem	O
,	O
roughly	O
speaking	O
it	O
states	O
that	O
every	O
sequence	O
of	O
equicontinuous	O
and	O
uniformly	B-Algorithm
bounded	O
maps	O
has	O
a	O
subsequence	O
that	O
converges	O
compactly	O
to	O
some	O
continuous	O
map	O
.	O
</s>
<s>
If	O
uniformly	B-Algorithm
,	O
then	O
compactly	O
.	O
</s>
<s>
If	O
is	O
a	O
compact	O
space	O
and	O
compactly	O
,	O
then	O
uniformly	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
locally	O
compact	O
space	O
,	O
then	O
compactly	O
if	O
and	O
only	O
if	O
locally	O
uniformly	B-Algorithm
.	O
</s>
