Page 22 - DMTH503_TOPOLOGY

Topology

Notes
Example 22: 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 23: Prove that every real number is a limit point of R – Q.
Solution: Let x be any real number, then 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.

1.6.3 Derived Set

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

Example 24: Every derived set in a topological space is a closed.
Solution: Let (X, T) be a topological space and A  X.
To show: D(A) is a closed set.

As we know that B is a closed set if D(B)  B.
Hence, D(A) is closed iff D[D(A)]  D(A).
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 – {x})  A  
 x  D(A)
Hence proved.
[For every nhd of an element of D(A) has at least one point of A].

Example 25: In any topological space, prove that A  D(A) is closed.
Solution: Let (X, T) be a topological space and A  X.

To prove: A  D(A) is a closed set.
Let x  X – A  D(A) be arbitrary then x  A  D(A) so that x  A, x  D(A)
x  D(A)   G  T with x  G s.t.

(G – {x})  A = 
 G  A =  ( x  A) …(1)

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