Page 45 - DMTH503_TOPOLOGY

Unit 3: The Order Topology

Proof: As we know Notes
B = {Finite intersections of S-sets}
S<
= S  {(a, b)| a, b  X, a < b} is a basis for the topology generated by the
sub-basis S .
<
If X has a smallest element a then (–, b) = [a , b) is open. If X has no smallest element, then the
o o
open ray (–, b) =  (a, c) is a union of open intervals and we do not need this open ray in the
a < c
basis. Similar remarks apply to the greatest element when it exists.
3.2 Summary

 Open interval : (a, b) = {x|a < x < b}

Closed interval : [a, b] = {x|a  x  b}
 Half open intervals : (a, b] = {x|a < x  b}

[a, b) = {x|a  x < b}
 The order topology T on the linearly ordered set X is the topology generated by all open
<
rays. A linearly ordered space is a linearly ordered set with the order topology.
 Open rays : (a, + ) = {x|x > a}
(–, a) = {x|x < a}

 Closed rays : (–, a] = {x|x  a}
[a, + ) = {x|x  a}

3.3 Keywords

Basis: A basis  for a topological space X with topology T is a collection of open sets in T such
that every open set in T can be written as a union of elements of .
Discrete Space: Let X be any non empty set and T be the collection of all subsets of X. Then T is
called the discrete topology on the set X. The topological space (X, T) is called a discrete space.
Open and Closed Set: Any set AT is called an open subset of X or simply a open set and X – A
is a closed subset of X.
3.4 Review Questions

1. Let X be an ordered set. If Y is a proper subset of X that is convex in X, does it follow that
Y is an interval or a ray in X?
2. Show that the dictionary order topology on the set R × R is the same as the product
topology R × R, where R denotes R in the discrete topology. Compare this topology with
d d
the standard topology on R .
2

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