Page 67 - DMTH503_TOPOLOGY

Unit 5: The Subspace Topology

Subset: If A and B are sets and every element of A is also an element of B, then A is subset of B Notes
denoted by A  B.
Subspace: Given a topological space (X, T) and a subset S of X, the subspace topology on S is
defined by
T = {S  :  T}
Topological Space: It is a set X together with T, a collection of subsets of X, satisfying the
following axioms. (1) The empty set and X are in T; (2) T is closed under arbitrary union and
(3) T is closed under finite intersection. Then collection T is called a topology on X.

5.4 Review Questions

1. Let X = {1, 2, 3, 4, 5}, A = {1, 2, 3}  X and
T = {, X, {1}, {2}, {1, 2}, {1, 4, 5}, {1, 2, 4, 5}}.
Find relative topology T, on A.

2. Let (X, T) be a topological space and X*  X. Let T* be the collection of all sets which are
intersections of X* with members of T. Prove that T* is a topology on X*.
3. Show that if Y is a subspace of X, and A  Y, then the topology A inherits as a subspace of
Y is the same as the topology it inherits as a subspace of X.
4. If T and T are topologies on X and T is strictly finer than T, what do you say about the
corresponding subspace topologies on the subset Y of X?

5. Let A be a subset of X. If  is a base for the topology of X, then the collection
 = {B  A : B  }
A
is a base for the subspace topology on A.

6. Let (Y, ) be a subspace of (X, T). If F and F are the collections of all closed subsets of (X, T)
1
and (Y, ) respectively, then F  F  Y  F.
1
5.5 Further Readings

Books Willard, Stephen. General Topology, Dover Publication (2004).
Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley
(1966).
Simmons. Introduction to Topology and Modern Analysis.
James & James. Mathematics Dictionary.

LOVELY PROFESSIONAL UNIVERSITY 61