Page 70 - DMTH503_TOPOLOGY

Topology

Notes Self Assessment

1. Show that if A is closed in Y and Y is closed in X, then A is closed in X.
2. Show that if A is closed in X and B is closed in Y, then A × B is closed in X × Y.

6.2 Limit Point

Let (X, T) be a topological space and A X. A point x  X is said to be the limit point or
accumulation point of A if each open set containing x contains at least one point of A different
from x.

Thus it is clear from the above definition that the limit point of a set A may or may not be the
point of A.

Note Limit point is also known as accumulation point or cluster point.

Example 4: Let X = {a, b, c} with topology T   , {a,b}, {c}, X  and A = {a}, then b is the
only limit point of A, because the open sets containing b namely {a, b} and X also contains a point
of A.
Where as ‘a’ and ‘b’ are not limit point of C = {c}, because the open set {a, b} containing these
points do not contain any point of C. The point ‘c’ is also not a limit point of C, since then open
set {c} containing ‘c’ does not contain any other point of C different from C. Thus, the set C = {c}
has no limit points.

Example 5: Prove that every real number is a limit point of R.
Solution: Let x  R
then every nhd of x contains at least one point of R other than x.

 x is a limit point of R.
But x was arbitrary.
 every real number is a limit point of R.

Example 6: Prove that every real number is a limit point of R  Q.
Solution: Let x be any real number, the every nhd of x contains at least one point of R – Q other
than x.

 x is a limit point of R – Q
But x was arbitrary
 every real number is a limit point of R – Q.

6.2.1 Derived Set

The set of all limit points of A is called the derived set of A and is denoted by D(A).

64 LOVELY PROFESSIONAL UNIVERSITY