Page 18 - DMTH503_TOPOLOGY

Topology

Notes so, N has the following properties
2
N = N  Y and N is a -nhd of y.
1 2 2
This completes the proof.
Step II: Conversely Let N be a T-nhd of y so that
2
 A  T s.t. y  A  N …(4)
2
To show: N  Y is a U-nhd of y.
2
 y  Y, y  A  y  Y  A [by (4)]
 y  A  Y  N  Y [by (3)]
2
A  T  A  Y 
Thus, we have y  A  Y  N  Y, where A  Y  .
2
This shows that N  Y is a -nhd of y.
2
Self Assessment

8. Let T = {X, , {p}, {p, q}, {p, q, t}, {p, q, r, s}, {p, r, s}} be the topology on
X = {p, q, r, s, t}

List the nhds of the points r, t.
9. Prove that a set G in a topological space X is open iff G is a nhd of each of its points.

1.5 Dense Set and Boundary Set

1.5.1 Dense Set and No where Dense

Let (X, T) be a topological space and A  X then A is said to be dense or everywhere dense in X
if A = X.

Example 13: Consider the set of rational number Q  R, then only closed set containing
Q in R, which shows that Q = R.
Hence, Q is dense in R.

Note Rational are dense in R and countable but irrational numbers are also dense in R
but not countable.

Example 14: Prove that A set is always dense in its subset
Solution: Let A  B then A  B  B
 A  B
 B  A

 B is dense in A.

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