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In
general topology In mathematics, general topology (or point set 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 differ ...
, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square) is a
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
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 coordinat ...
''S'', i.e. on the set of points (''x'',''y'') in the plane such that and


Construction

The
lexicographical order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
ing 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 specific 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, ...
induced by this ordering.


Properties

The order topology makes ''S'' into a
completely normal Completely may refer to: * ''Completely'' (Diamond Rio album) * ''Completely'' (Christian Bautista album), 2005 * "Completely", a song by American singer and songwriter Michael Bolton * "Completely", a song by Serial Joe from ''(Last Chance) A ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
. 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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
. At the same time, ''S'' contains an
uncountable In mathematics, an uncountable set, informally, 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 number is larger tha ...
number of
pairwise disjoint In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
open intervals, each
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
, 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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
(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 In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
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


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