<s>
Any	O
set	O
of	O
cardinal	O
numbers	O
or	O
ordinal	B-Language
numbers	I-Language
(	O
more	O
strongly	O
,	O
these	O
are	O
well-orders	O
)	O
.	O
</s>
<s>
The	O
real	O
numbers	O
form	O
an	O
initial	O
unbounded	O
totally	O
ordered	O
set	O
that	O
is	O
connected	O
in	O
the	O
order	B-Algorithm
topology	I-Algorithm
(	O
defined	O
below	O
)	O
.	O
</s>
<s>
For	O
example	O
,	O
an	O
order	O
is	O
well	B-Algorithm
founded	I-Algorithm
if	O
it	O
has	O
the	O
descending	O
chain	O
condition	O
.	O
</s>
<s>
Either	O
by	O
direct	O
proof	O
or	O
by	O
observing	O
that	O
every	O
well	O
order	O
is	O
order	O
isomorphic	O
to	O
an	O
ordinal	B-Language
one	O
may	O
show	O
that	O
every	O
finite	O
total	O
order	O
is	O
order	O
isomorphic	O
to	O
an	O
initial	O
segment	O
of	O
the	O
natural	O
numbers	O
ordered	O
by	O
<	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
total	O
order	O
on	O
a	O
set	O
with	O
k	O
elements	O
induces	O
a	O
bijection	B-Algorithm
with	O
the	O
first	O
k	O
natural	O
numbers	O
.	O
</s>
<s>
A	O
bijective	B-Algorithm
map	I-Algorithm
between	O
two	O
totally	O
ordered	O
sets	O
that	O
respects	O
the	O
two	O
orders	O
is	O
an	O
isomorphism	O
in	O
this	O
category	O
.	O
</s>
<s>
We	O
can	O
use	O
these	O
open	O
intervals	O
to	O
define	O
a	O
topology	B-Architecture
on	O
any	O
ordered	O
set	O
,	O
the	O
order	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
When	O
more	O
than	O
one	O
order	O
is	O
being	O
used	O
on	O
a	O
set	O
one	O
talks	O
about	O
the	O
order	B-Algorithm
topology	I-Algorithm
induced	O
by	O
a	O
particular	O
order	O
.	O
</s>
<s>
For	O
instance	O
if	O
N	O
is	O
the	O
natural	O
numbers	O
,	O
is	O
less	O
than	O
and	O
greater	O
than	O
we	O
might	O
refer	O
to	O
the	O
order	B-Algorithm
topology	I-Algorithm
on	O
N	O
induced	O
by	O
and	O
the	O
order	B-Algorithm
topology	I-Algorithm
on	O
N	O
induced	O
by	O
(	O
in	O
this	O
case	O
they	O
happen	O
to	O
be	O
identical	O
but	O
will	O
not	O
in	O
general	O
)	O
.	O
</s>
<s>
The	O
order	B-Algorithm
topology	I-Algorithm
induced	O
by	O
a	O
total	O
order	O
may	O
be	O
shown	O
to	O
be	O
hereditarily	O
normal	O
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
results	O
relating	O
properties	O
of	O
the	O
order	B-Algorithm
topology	I-Algorithm
to	O
the	O
completeness	O
of	O
X	O
:	O
</s>
<s>
If	O
the	O
order	B-Algorithm
topology	I-Algorithm
on	O
X	O
is	O
connected	O
,	O
X	O
is	O
complete	O
.	O
</s>
<s>
X	O
is	O
connected	O
under	O
the	O
order	B-Algorithm
topology	I-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
complete	O
and	O
there	O
is	O
no	O
gap	O
in	O
X	O
(	O
a	O
gap	O
is	O
two	O
points	O
a	O
and	O
b	O
in	O
X	O
with	O
a	O
<	O
b	O
such	O
that	O
no	O
c	O
satisfies	O
a	O
<	O
c	O
<	O
b	O
.	O
)	O
</s>
<s>
X	O
is	O
complete	O
if	O
and	O
only	O
if	O
every	O
bounded	O
set	O
that	O
is	O
closed	O
in	O
the	O
order	B-Algorithm
topology	I-Algorithm
is	O
compact	O
.	O
</s>
<s>
A	O
totally	O
ordered	O
set	O
(	O
with	O
its	O
order	B-Algorithm
topology	I-Algorithm
)	O
which	O
is	O
a	O
complete	O
lattice	O
is	O
compact	O
.	O
</s>
