Definition. Let (X, J) be a topological space. For Y C X, the collection Jy = {U | U = Vn Y for some V E J} is a topology on Y called the subspace topology. It is also called the relative topology

Definition. Let (X, J) be a topological space. For Y C X, the collection
Jy = {U | U = Vn Y for some V E J}
is a topology on Y called the subspace topology. It is also called the relative topology
on Y inherited from X. The space (Y, Jy) is called a (topological) subspace of X. If
U E Jy we say U is open in Y.
Definition. Let X be a set totally ordered by <. Let B be the collection of all subsets of
X that are any of the following forms:
{xXx<a} or {xEX|a<x} or {xEX|a<x<b}
for a, b E X. Then B is a basis for a topology J, called the order topology on X.
Definition. Given sets A and B, their product (or Cartesian product) A X B is the set
of all ordered pairs (a, b) such that a E A and b E B. If A and B are totally ordered
by <A and <B, respectively, then the dictionary order or lexicographic order < on
A X B is specified by defining (a1, b1) < (a2, b2) if al <A d2, or if a1 = a2 and b1 <B b2.
Example. The square [0, 1] x [0, 1] with the lexicographic order and its associated order
topology is called the lexicographically ordered square.
Exercise 3.31. Consider the following subspaces of the lexicographically ordered square:
(1) D = {(x, 7) 10 <x<1}
(2) E = {(. >) 10 <><1}.
(3) F = {(x, 1) |0 <x<1}.
As sets they are all lines. Describe their relative topologies, especially noting any connec-
tions to topologies you have seen already.

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