Page 71 - DMTH503_TOPOLOGY

Unit 6: Closed Sets and Limit Point

Notes

Notes 1. In terms of derived set, the closure of a set A X is defined as A = A + D(A)
= A  D(A).
2. If every point of A is an isolated point of A, then A is known as isolated set.

Example 7: Every derived set in a topological space is a closed.
Solution: Let (X, T) be a topological space and A  X.

Aim: D(A) is a closed set.
Recall that B is a closed set if D(B)  B.

Hence D(A) is closed iff D(A)  D(A).
D
Let x D D(A)   be arbitrary, then x is a limit point of D(A) so that G {x}   D(A)   G T

with x  G.
 G    A    x D(A).
x

Hence the result.
[For every nhd of an element of D(T) has at least one point of A].

Example 8: Let (X, T) be a topological space and A  X, then A is closed iff A’  A or
A  D(A).

Solution: Let A be closed.
c
 A is open.
Let x  A .
c
c
Then A is an open set containing x but containing no point of A other than x.
This shows that x is not a limit point of A.
Thus, no point of A is a limit point of A.
c
Consequently, every limit point of A is in A and therefore
A  A

Conversely, Let A  A
we have to show that A is closed.
Let x be arbitrary point of A . c
Then x  A c

 x  A
 x  A and x  A
 x  A and x not a limit point of A.
  an open set G such that x  G and G  A = 

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