Page 65 - DMTH503_TOPOLOGY

Unit 5: The Subspace Topology

Let G  be arbitrary, then G  T, by (1). Notes
We can write G = H  Y for some T-open set H.
Now, G = H  Y, G  T  H  Y  T

Again H  Y  T, H  T and Y  X  Y  T
Conversely, let (Y, )  (X, T) and let Y  T for any G   G  T
 G    A  T s.t. G = A  Y

Again A  T, Y  T  A  Y  T  G  T
Finally, any G   G  T.

5.1.2 Basis for the Subspace Topology

Example 8: Consider the subset Y = [0, 1] of the real line , in the subspace topology. The
subspace topology has as basis all sets of the form (a, b)  Y, where (a, b) is an open interval in
, such a set is of one of the following types:
ì (a,b) if a and b are in Y,
ï
ï [0,b) if only b is in Y,
(a, b)  Y = í
ï (a,1] if only a is in Y,
ï Y or  if neither a nor b is in Y.
î
By definition, each of these sets is open in Y. But sets of the second and third types are not open
in the larger space .
Note that these sets form a basis for the order topology on Y. Thus, we see that in the case of the
set Y = [0, 1], its subspace topology (as a subspace of ) and its order topology are the same.

Example 9: Let Y be the subset [0, 1)  {2} of . In the subspace topology on Y the one-
5
3
point set {2} is open, because it is the intersection of the open set ( , ) with Y. But in the order
2 2
topology on Y, the set {2} is not open. Any basis element for the order topology on Y that
contains 2 is of the form
{x  x  Y and a < x  2}
for some a  Y; such a set necessarily contains points of Y less than 2.
Lemma 1: If  is a basis for the topology of X, then the collection  = {B  Y : B  } is a basis for
Y
the subspace topology on Y.
Proof: Given -open in X and given y   Y, we can choose an element B of  such that
y  B  . Then y  B  Y   Y
Now as we know

“If X is a topological space and  is a collection of open sets of X such that for each open set of
X and each x in , there is an element c of  such that x    . The  is a basis for the topology
of X.”
Thus, we can say that  is a basis for the subspace topology on Y.
Y

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