Page 64 - DMTH503_TOPOLOGY

Topology

Notes Theorem 2: Let (Y, ) be a subspace of a topological space (X, T). A subset of Y is -nhd. of a point
y  Y iff it is the intersection of Y with a T-nhd. of the point y  Y.
Proof: Let (Y, )  (X, T) and y  Y be arbitrary, then y  X.

(I) Let N be a -nhd of y, then
1
 V  s.t. y  V  N …(1)
1
To prove : N = N  Y for some T-nhd N of y.
1 2 2
y  V    G  T s.t. V = G  Y

 y  G  Y  y  G, y  Y …(2)
Write N = N  G.
2 1
Then N  N , G  N .
1 2 2
so, (2) implies y  G  N , where G  T
2
This shows that N is a T-nhd of y.
2
N  Y = (N  G)  Y
2 1
= (N  Y)  (G  Y)
1
= (N  Y)  V
1
= N  V [by (1)]
1
= N N  Y and V  N
1 1 1
Finally, N has the following properties
2
N = N  Y and N is a T-nhd of y.
1 2 2
This completes the proof.
(II) Conversely, Let N be a T-nhd. of y so that
2
 A  T s.t. y  A  N …(3)
2
We are to prove that N  Y is a -nhd of y.
2
 y  Y, y  A  y  Y  A [by (3)]

 y  A  Y  N  Y [by (3)]
2
A  T  A  Y 
Thus, we have y  A  Y  N  Y, where A  Y  .
2
 N  Y is a -nhd of y.
2

Example 7: Let (Y, ) be a subspace of a topological space (X, T). Then every -open set
is also T-open iff Y is T-open.
Solution: Let (Y, )  (X, T) and let
any G   G  T …(1)
i.e. every -open set is also T-open set.

To show: Y is T-open, it is enough to prove that y  T.

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