Page 93 - DMTH503_TOPOLOGY

Unit 9: The Metric Topology

Open Set Notes

Let (X,) be a metric space.
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A non-empty set GX is called an open set if any xGrR s.t. S (x)G.
r
Limit Point

Let (X,) be a metric space and AX. A point xX is called a limit point or limiting point or
accumulation point or cluster point if every open sphere centered on x contains a point of A
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other than x, i.e., xX is called limit point of A if (S – {x})A, rR .
r(x)
The set of all limiting points of a set A is called derived set of A and is denoted by D(A).
Closed Set

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.
9.1.5 Convergence of a Sequence in a Metric Space

Let <x > be a sequence in a metric space (X, ). This sequence is said to coverage to x X, if given
n 0
any  > 0,  n N s.t. n n (x , x ) < or equivalently, given any  > 0, n N s.t. n n
o 0 n 0 0 0
x S (x ).
n  0
9.1.6 Theorems on Closed Sets and Open Sets

Theorem 1: In a metric space (X, ) ,  and X are closed sets.
Proof: Let (X, ) be a metric space.

To prove that and X are closed sets.
D () = 
 D ()  .
is a closed set.

All the limiting points of X belong to X. For X is the universal set.
i.e., any x D (x)  x X
D (X)  X
X is a closed set.

Theorem 2: Let (X, d) be a metric space. Show that F X, F is closed Fis open.
Proof: Let (X, d) be a metric space.
Let F be a closed subset of X, so that D(F) F.
To prove that F is open in X.

Let x  Fbe arbitrary. Then x F.
D (F) F, x F  x D (F)
 (S – {x}) F = for some r > 0
r(x)

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