Page 66 - DMTH503_TOPOLOGY

Topology

Notes Lemma 2: Let Y be a subspace of X. If is open in Y and Y is open in X, then is open in X.
Proof: Since is open in Y,
= Y  V for some set V open in X.
Since Y and V are both open in X,
so is Y  V.

5.1.3 Subspace of Product Topology

Theorem 3: If A is a subspace of X and B is a subspace of Y, then the product topology on A  B is
the same as the topology A  B inherits as a subspace of X  Y.
Proof: The set  V is the general basis element for X  Y, where is open in X and V is open
in Y.
, (  V)  (A  B) is the general basis element for the subspace topology on A  B.
Now, (  V)  (A  B) = (  A)  (V  B).
Since  A and V  B are the general open sets for the subspace topologies on A and B,
respectively, the set (  A)  (V  B) is the general basis element for the product topology on
A  B.
So, we can say that the bases for the subspace topology on A  B and for the product topology on
A × B are the same.
Hence, the topologies are the same.

5.2 Summary

 A subspace of a topological space is itself a topology space.

 If  is a basis for the topology of X, then the collection  = {B  Y : B  } is a basis for the
Y
subspace topology on Y.
 Let Y be a subspace of X. If is open in Y and Y is open in X, then is open in X.

 If A is a subspace of X and B is a subspace of Y then the product topology on A  B is the
same as the topology A  B inheritsas a subspace of X  Y.

5.3 Keywords

Basis: Let X be a topological space A set  of open set is called a basis for the topology if every
open set is a union of sets in .
Closed Set: Let (X, T) be a topological space. Let set A  T. Then X–A is a closed set.
Intersection: The intersection of A and B is written A  B. x  A  B  x  A and x  B.

Neighborhood: Let (X, T) be a topological space. A  X is called a neighborhood of a point x  X
if  G  T with x  G s.t. G  A.
Open set: Let (X, T) be a topological space. Any set A  T is called an open set.

Product Topology: Let X and Y be topological space. The product topology on X  Y is the
topology having as basis the collection  of all sets of the form  V, where is an open subset
of X and V is an open subset of Y.

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