<s>
In	O
mathematics	O
,	O
in	O
the	O
theory	O
of	O
functions	B-Architecture
of	I-Architecture
several	I-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
is	O
a	O
domain	O
which	O
is	O
maximal	O
in	O
the	O
sense	O
that	O
there	O
exists	O
a	O
holomorphic	O
function	O
on	O
this	O
domain	O
which	O
cannot	O
be	O
extended	B-Algorithm
to	O
a	O
bigger	O
domain	O
.	O
</s>
<s>
In	O
the	O
case	O
,	O
every	O
open	O
set	O
is	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
:	O
we	O
can	O
define	O
a	O
holomorphic	O
function	O
with	O
zeros	O
accumulating	O
everywhere	O
on	O
the	O
boundary	O
of	O
the	O
domain	O
,	O
which	O
must	O
then	O
be	O
a	O
natural	O
boundary	O
for	O
a	O
domain	O
of	O
definition	O
of	O
its	O
reciprocal	O
.	O
</s>
<s>
For	O
this	O
is	O
no	O
longer	O
true	O
,	O
as	O
it	O
follows	O
from	O
Hartogs	B-Algorithm
 '	I-Algorithm
lemma	I-Algorithm
.	O
</s>
<s>
If	O
are	O
domains	O
of	O
holomorphy	O
,	O
then	O
their	O
intersection	O
is	O
also	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
If	O
is	O
an	O
ascending	O
sequence	O
of	O
domains	O
of	O
holomorphy	O
,	O
then	O
their	O
union	O
is	O
also	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
(	O
see	O
Behnke-Stein	O
theorem	O
)	O
.	O
</s>
<s>
If	O
and	O
are	O
domains	O
of	O
holomorphy	O
,	O
then	O
is	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
The	O
first	O
Cousin	O
problem	O
is	O
always	O
solvable	O
in	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
;	O
this	O
is	O
also	O
true	O
,	O
with	O
additional	O
topological	O
assumptions	O
,	O
for	O
the	O
second	O
Cousin	O
problem	O
.	O
</s>
