Page 16 - DMTH503_TOPOLOGY

Topology

Notes 6. On the real line show that every open interval is an open but every open set need not be
an open interval.
7. Let (Y, U) be a subspace of a topological space (X, T). Then every U-open set is also T-open
iff Y is T-open.

1.4 Neighborhood

Let (X, T) be a topological space. A  X is called a neighbourhood of a point x  X if  G  T with
x  G such that G  A. The word neighborhood is, in short, written as ‘nhd’.
Let G be any open set such that G  X with x  G is also nhd of a point x  X.

Example 12: Let T = {, X, {b}, {a, b}, {a, b, d}}, be a topology on X = {a, b, c, d}. Find T-nhds
of (i) a, (ii) b and (iii) c.

Solution:
(i) T-open sets containing ‘a’ are X, {a, b}, {a, b, d}.
super set of X is X

supersets of {a, b} are {a, b}, {a, b, c}, {a, b, d}, X
supersets of {a, b, d} are {a, b, d}, X.
T-nhds of ‘a’ are {a, b}, {a, b, c}, {a, b, d}, X
(ii) T-open sets containing b are
{b}, {a, b}, {a, b, d}, X

supersets of {a, b} are {a, b}, {a, b, c}, {a, b, d}, X
supersets of {a, b, d} are {a, b, d}, X
supersets of {b} are {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {b, c, d}, {a, b, d}, X

T-nhds of ‘b’ are {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}, X
(iii) T-open set containing ‘c’ is X.
Hence T-nhd of ‘c’ is X.
Theorem 7: Let (X, T) be a topological space and A  X. Then A is T-open  A contains T-nhds of
each of its points.
Proof: Let (X, T) be a topological space and A  X.
Step I: Given A is an open set.
To show: A contains T-nhd of each of its points. Clearly x  A  A  x  A and A is an open set.
This shows that A contains T-nhd of each of its points.
Step II: Given A contains T-nhd of each of its point, then any x  A   nhd N  X such that
x
x  N  A …(1)
x
To show: A is an open set
By definition of nhd,  open set G s.t.
x
x  G  N …(2)
x x

10 LOVELY PROFESSIONAL UNIVERSITY