Lexicographic Order Topology On The Unit Square
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In
general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometri ...
, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square) is a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on the
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordin ...
''S'', i.e. on the set of points (''x'',''y'') in the
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
such that and


Construction

The lexicographical ordering gives a total ordering \prec on the points in the unit square: if (''x'',''y'') and (''u'',''v'') are two points in the square, if and only if either or both and . Stated symbolically, (x,y)\prec (u,v)\iff (x The lexicographic order topology on the unit square is the
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...
induced by this ordering.


Properties

The order topology makes ''S'' into a
completely normal In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
. Since the lexicographical order on ''S'' can be proven to be complete, this topology makes ''S'' into a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
. At the same time, ''S'' contains an
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
number of
pairwise disjoint In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A c ...
open intervals, each homeomorphic to the real line, for example the intervals U_x=\ for 0\le x\le 1. So ''S'' is not separable, since any dense subset has to contain at least one point in each U_x. Hence ''S'' is not
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
(since any compact metric space is separable); however, it is
first countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
. 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 The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, ...
*
Long line Long line or longline may refer to: *'' Long Line'', an album by Peter Wolf * Long line (topology), or Alexandroff line, a topological space *Long line (telecommunications), a transmission line in a long-distance communications network *Longline fi ...


Notes


References

* {{Citation, first=L. A., last=Steen, first2=J. A., last2=Seebach, title=Counterexamples in Topology, publisher=Dover, year=1995, isbn=0-486-68735-X, title-link=Counterexamples in Topology General topology Topological spaces