<s>
In	O
mathematics	O
,	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
is	O
a	O
topology	O
defined	O
on	O
the	O
set	O
of	O
continuous	O
maps	O
between	O
two	O
topological	O
spaces	O
.	O
</s>
<s>
The	O
compact-open	B-Algorithm
topology	I-Algorithm
is	O
one	O
of	O
the	O
commonly	O
used	O
topologies	O
on	O
function	B-Algorithm
spaces	I-Algorithm
,	O
and	O
is	O
applied	O
in	O
homotopy	O
theory	O
and	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
If	O
the	O
codomain	B-Algorithm
of	O
the	O
functions	O
under	O
consideration	O
has	O
a	O
uniform	O
structure	O
or	O
a	O
metric	O
structure	O
then	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
is	O
the	O
"	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
compact	O
sets.	O
"	O
</s>
<s>
That	O
is	O
to	O
say	O
,	O
a	O
sequence	O
of	O
functions	O
converges	B-Algorithm
in	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
precisely	O
when	O
it	O
converges	B-Algorithm
uniformly	I-Algorithm
on	O
every	O
compact	O
subset	O
of	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
Then	O
the	O
collection	O
of	O
all	O
such	O
is	O
a	O
subbase	O
for	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
on	O
.	O
</s>
<s>
However	O
,	O
the	O
modified	O
definition	O
is	O
crucial	O
if	O
one	O
wants	O
the	O
convenient	O
category	O
of	O
compactly	O
generated	O
weak	O
Hausdorff	O
spaces	O
to	O
be	O
Cartesian	B-Application
closed	I-Application
,	O
among	O
other	O
useful	O
properties	O
.	O
</s>
<s>
This	O
adjoint	O
coincides	O
with	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
and	O
may	O
be	O
used	O
to	O
uniquely	O
define	O
it	O
.	O
</s>
<s>
If	O
is	O
a	O
one-point	O
space	O
then	O
one	O
can	O
identify	O
with	O
,	O
and	O
under	O
this	O
identification	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
agrees	O
with	O
the	O
topology	O
on	O
.	O
</s>
<s>
More	O
generally	O
,	O
if	O
is	O
a	O
discrete	O
space	O
,	O
then	O
can	O
be	O
identified	O
with	O
the	O
cartesian	O
product	O
of	O
copies	O
of	O
and	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
agrees	O
with	O
the	O
product	O
topology	O
.	O
</s>
<s>
If	O
is	O
,	O
,	O
Hausdorff	O
,	O
regular	O
,	O
or	O
Tychonoff	O
,	O
then	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
has	O
the	O
corresponding	O
separation	O
axiom	O
.	O
</s>
<s>
If	O
is	O
Hausdorff	O
and	O
is	O
a	O
subbase	O
for	O
,	O
then	O
the	O
collection	O
is	O
a	O
subbase	O
for	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
on	O
.	O
</s>
<s>
If	O
is	O
a	O
metric	O
space	O
(	O
or	O
more	O
generally	O
,	O
a	O
uniform	O
space	O
)	O
,	O
then	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
is	O
equal	O
to	O
the	O
topology	B-Algorithm
of	I-Algorithm
compact	I-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
if	O
is	O
a	O
metric	O
space	O
,	O
then	O
a	O
sequence	O
converges	B-Algorithm
to	O
in	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
if	O
and	O
only	O
if	O
for	O
every	O
compact	O
subset	O
of	O
,	O
converges	B-Algorithm
uniformly	I-Algorithm
to	O
on	O
.	O
</s>
<s>
If	O
is	O
compact	O
and	O
is	O
a	O
uniform	O
space	O
,	O
then	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
is	O
equal	O
to	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
topological	O
spaces	O
,	O
with	O
locally	O
compact	O
Hausdorff	O
(	O
or	O
even	O
just	O
locally	O
compact	O
preregular	O
)	O
,	O
then	O
the	O
composition	B-Application
map	I-Application
given	O
by	O
is	O
continuous	O
(	O
here	O
all	O
the	O
function	B-Algorithm
spaces	I-Algorithm
are	O
given	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
and	O
is	O
given	O
the	O
product	O
topology	O
)	O
.	O
</s>
<s>
If	O
is	O
compact	O
,	O
and	O
is	O
a	O
metric	O
space	O
with	O
metric	O
,	O
then	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
on	O
is	O
metrisable	O
,	O
and	O
a	O
metric	O
for	O
it	O
is	O
given	O
by	O
for	O
in	O
.	O
</s>
<s>
The	O
compact	B-Algorithm
open	I-Algorithm
topology	I-Algorithm
can	O
be	O
used	O
to	O
topologize	O
the	O
following	O
sets	O
:	O
</s>
