- Source: Lexicographic order topology on the unit square
In general topology, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square) is a topology on the unit square S, i.e. on the set of points (x,y) in the plane such that 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
Construction
The lexicographical ordering gives a total ordering
≺
{\displaystyle \prec }
on the points in the unit square: if (x,y) and (u,v) are two points in the square, (x,y)
≺
{\displaystyle \scriptstyle \prec }
(u,v) if and only if either x < u or both x = u and y < v. Stated symbolically,
(
x
,
y
)
≺
(
u
,
v
)
⟺
(
x
<
u
)
∨
(
x
=
u
∧
y
<
v
)
{\displaystyle (x,y)\prec (u,v)\iff (x
The lexicographic order topology on the unit square is the order topology induced by this ordering.
Properties
The order topology makes S into a completely normal Hausdorff space. Since the lexicographical order on S can be proven to be complete, this topology makes S into a compact space. At the same time, S contains an uncountable number of pairwise disjoint open intervals, each homeomorphic to the real line, for example the intervals
U
x
=
{
(
x
,
y
)
:
1
/
4
<
y
<
1
/
2
}
{\displaystyle U_{x}=\{(x,y):1/4
for
0
≤
x
≤
1
{\displaystyle 0\leq x\leq 1}
. So S is not separable, since any dense subset has to contain at least one point in each
U
x
{\displaystyle U_{x}}
. Hence S is not metrizable (since any compact metric space is separable); however, it is first countable. Also, S is connected and locally connected, but not path connected and not locally path connected. Its fundamental group is trivial.
See also
List of topologies
Long line
Notes
References
Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X
Kata Kunci Pencarian:
- Lexicographic order topology on the unit square
- Lexicographic order
- Long line (topology)
- Counterexamples in Topology
- List of topologies
- Locally connected space
- Ordered field
- Cyclic order
- Gδ space
- Linear continuum
