Page 104 - DMTH503_TOPOLOGY

Topology

Notes  Let (X, ) be a metric space. A non empty set G X is called an open set if any x G 
+
r R s.t. S G.
r(x)
 Let (X, ) be a metric space and A X. A point x X is called a limit point if every open
sphere centered on x contains a point of A other than x, i.e. x X is called the limit point

S  {x}   A   , r  R .
of A if  r
(x)
 Let (X, ) be a metric space and A X. A is called a closed set if the derived set of A i.e.
D (A) A i.e. if every limit point of A belongs to the set itself.
 Set <x > be a sequence in a metric space (X, ). This sequence is said to converge to x X,
n 0
if given any > 0, n N s.t. n n (x , x ) < .
o 0 n 0
+
 A point x A is called an interior point of A if r R s.t. S  A.
r
(x)
 The closure of A, denoted by A , is defined as the intersection of all closed sets that
contain A.
 Boundary of a set A is denoted by b(A) is defined as b(A) = X – A°(X – A)°.
 The exterior of A is defined as the set (X – A)° and is denoted by ext (A).
 A is said to be dense or everywhere dense in X if A = X.
 A is said to be nowhere dense if (A)   .

 A metric space (X, d) is said to be separable if A X s.t. A is countable and A  X.
 A sequence defined on a metric space (X, d) is said to be uniformly convergent if given
 > 0, n N s.t. n n
0 0
d (f (x), f(x)) <   x X.
n
9.3 Keywords

Frechet Space: A topology space (X, T) is said to satisfy the T – axiom of separation if given a pair
1
of distinct point x, y X.
G, H T s.t. x  G, y G; y H, x H.
In this case the space (X, T) is called Frechet Space.

Intersection: The intersection of two sets A and B, denoted by A B, is defined as the set
containing those elements which belong to A and B both. Symbolically
A B = {x : x A and x B}

Union: The union of two sets A ad B, denoted by A B, is defined as the set of those elements
which either belong to A or to B. Symbolically
A B = {x : x A or x : B}

9.4 Review Questions

1. In any metric space (X, d), show that

(a) an arbitrary intersection of closed sets is closed.
(b) any finite union of closed sets is closed.

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