Page 120 - DMTH503_TOPOLOGY

Topology

Notes
Example 6: The spaces  and C are connected.
n
n
n
Solution: We know that  is a topological space can be regarded as the product of n replicas of
n
the real line . But  is connected therefore  is connected since the product of any non-empty
class of connected spaces is connected.
n
2n
We next prove that C and  are essentially the same as topological spaces by taking a
n
homomorphism f of C onto  .
2n
n
Let z = (z , z , …, z ) be an arbitrary element in C .
1 2 n
Let us suppose that each coordinate z is of the form
K
z = a + ib
K K K
where a and b are its real and imaginary parts.
K K
Let us define f by
f (z) = (a , b , a , b , …, a , b ).
1 1 2 2 n n
f is clearly a one-to-one mapping of C onto  and if we observe that ||f (z)|| = ||z||, then f is a
n
2n
2n
n
homeomorphism which shows that  is connected. Hence C is also connected.
Self Assessment
4. Show that if f is continuous map of a connected space X into R, then f(X) is an interval.
5. Show that a subset  of the real line that contains at least two distinct points is connected
if and only if it is an interval.

11.3 Summary

 A topological space X is said to be connected if it cannot be written as the union of two
disjoint non-empty open sets.

 The closure of a connected set is connected.
 If every two points of a set E are contained in some connected subset of E, then E is
connected.

 A continuous image of connected space is connected.
 The set of real numbers with the usual metric is a connected space.
 A subspace of the real line R is connected iff it is an interval. In particular, R is connected.

11.4 Keyword

Separated set: Let A, B be subsets of a topological space (X, T). Then the set A and B are said to be
separated iff
(i) A  , B  

(ii) A B = , A B = 

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