<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
an	O
almost	B-Algorithm
open	I-Algorithm
map	I-Algorithm
between	O
topological	O
spaces	O
is	O
a	O
map	O
that	O
satisfies	O
a	O
condition	O
similar	O
to	O
,	O
but	O
weaker	O
than	O
,	O
the	O
condition	O
of	O
being	O
an	O
open	O
map	O
.	O
</s>
<s>
As	O
described	O
below	O
,	O
for	O
certain	O
broad	O
categories	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
surjective	B-Algorithm
linear	O
operators	O
are	O
necessarily	O
almost	O
open	O
.	O
</s>
<s>
Given	O
a	O
surjective	B-Algorithm
map	I-Algorithm
a	O
point	O
is	O
called	O
a	O
for	O
and	O
is	O
said	O
to	O
be	O
(	O
or	O
)	O
if	O
for	O
every	O
open	O
neighborhood	O
of	O
is	O
a	O
neighborhood	O
of	O
in	O
(	O
note	O
that	O
the	O
neighborhood	O
is	O
not	O
required	O
to	O
be	O
an	O
neighborhood	O
)	O
.	O
</s>
<s>
A	O
surjective	B-Algorithm
map	I-Algorithm
is	O
called	O
an	O
if	O
it	O
is	O
open	O
at	O
every	O
point	O
of	O
its	O
domain	O
,	O
while	O
it	O
is	O
called	O
an	O
each	O
of	O
its	O
fibers	B-Algorithm
has	O
some	O
point	O
of	O
openness	O
.	O
</s>
<s>
A	O
linear	O
map	O
between	O
two	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
is	O
called	O
a	O
or	O
an	O
if	O
for	O
any	O
neighborhood	O
of	O
in	O
the	O
closure	O
of	O
in	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Importantly	O
,	O
some	O
authors	O
use	O
a	O
different	O
definition	O
of	O
"	O
almost	B-Algorithm
open	I-Algorithm
map	I-Algorithm
"	O
in	O
which	O
they	O
instead	O
require	O
that	O
the	O
linear	O
map	O
satisfy	O
:	O
for	O
any	O
neighborhood	O
of	O
in	O
the	O
closure	O
of	O
in	O
(	O
rather	O
than	O
in	O
)	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
;	O
</s>
<s>
If	O
a	O
linear	O
map	O
is	O
almost	O
open	O
then	O
because	O
is	O
a	O
vector	O
subspace	O
of	O
that	O
contains	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
the	O
map	O
is	O
necessarily	O
surjective	B-Algorithm
.	O
</s>
<s>
For	O
this	O
reason	O
many	O
authors	O
require	O
surjectivity	B-Algorithm
as	O
part	O
of	O
the	O
definition	O
of	O
"	O
almost	O
open	O
"	O
.	O
</s>
<s>
Every	O
surjective	B-Algorithm
open	O
map	O
is	O
an	O
almost	B-Algorithm
open	I-Algorithm
map	I-Algorithm
but	O
in	O
general	O
,	O
the	O
converse	O
is	O
not	O
necessarily	O
true	O
.	O
</s>
<s>
If	O
a	O
surjection	B-Algorithm
is	O
an	O
almost	B-Algorithm
open	I-Algorithm
map	I-Algorithm
then	O
it	O
will	O
be	O
an	O
open	O
map	O
if	O
it	O
satisfies	O
the	O
following	O
condition	O
(	O
a	O
condition	O
that	O
does	O
depend	O
in	O
any	O
way	O
on	O
'	O
s	O
topology	O
)	O
:	O
</s>
<s>
That	O
is	O
,	O
if	O
is	O
a	O
continuous	O
surjection	B-Algorithm
then	O
it	O
is	O
an	O
open	O
map	O
if	O
and	O
only	O
if	O
it	O
is	O
almost	O
open	O
and	O
it	O
satisfies	O
the	O
above	O
condition	O
.	O
</s>
<s>
Theorem	O
:	O
If	O
is	O
a	O
surjective	B-Algorithm
linear	O
operator	O
from	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
onto	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
then	O
is	O
almost	O
open	O
.	O
</s>
<s>
Theorem	O
:	O
If	O
is	O
a	O
surjective	B-Algorithm
linear	O
operator	O
from	O
a	O
TVS	O
onto	O
a	O
Baire	O
space	O
then	O
is	O
almost	O
open	O
.	O
</s>
<s>
The	O
two	O
theorems	O
above	O
do	O
require	O
the	O
surjective	B-Algorithm
linear	O
map	O
to	O
satisfy	O
topological	O
conditions	O
.	O
</s>
<s>
Theorem	O
:	O
If	O
is	O
a	O
complete	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
,	O
is	O
a	O
Hausdorff	O
TVS	O
,	O
and	O
is	O
a	O
closed	O
and	O
almost	O
open	O
linear	O
surjection	B-Algorithm
,	O
then	O
is	O
an	O
open	O
map	O
.	O
</s>
<s>
Theorem	O
:	O
Suppose	O
is	O
a	O
continuous	O
linear	O
operator	O
from	O
a	O
complete	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
into	O
a	O
Hausdorff	O
TVS	O
If	O
the	O
image	O
of	O
is	O
non-meager	O
in	O
then	O
is	O
a	O
surjective	B-Algorithm
open	O
map	O
and	O
is	O
a	O
complete	O
metrizable	O
space	O
.	O
</s>
