Page 113 - DMTH503_TOPOLOGY

Unit 11: Connected Spaces, Connected Subspaces of Real Line

Notes

Note A set is said to be connected iff it has no separation

Example 1:
(1) Let X be an indiscrete topological space. Then X is connected since the indiscrete topology
consists of the empty set  and the whole space X only.
(2) Let X be a discrete topological space with at least two elements. Then X is disconnected
c
since if A is any non-empty proper subset of X, then A and A are disjoint non-empty open
subsets of X such that X = A A . c
(3)  is connected. Since  cannot be expressed as the union of two non-empty separated sets.
So  has no separation and is therefore connected.
Theorem 1: In a topological space X the following statements are equivalent:
(i) X is connected;

(ii) The empty set  and the whose space X are the only subsets of X that are both open and
closed in X i.e. X has no non-trivial subset that is both open and closed in X;
(iii) X cannot be represented as the union of two non-empty disjoint closed sets.

(iv) X cannot be represented as the union of two non-empty separated sets.
Proof: We shall prove the theorem by showing that
(i)  (ii)  (iii)  (iv)  (i)
(i)  (ii)

Let X be connected.
Suppose A is a non-trivial subset of X that is simultaneously open and closed in X. Then B = A is
c
non-empty, open and X = A B, A B = 
This is contrary to the given hypothesis that X is connected and accordingly (ii) must be true
(ii)  (iii)
Let (ii) be true.
Suppose X = A U B, where A and B are two disjoint non-empty closed sets.

c
Then A = B is a non-trivial subset of X that is open as well as closed in X. This contradicts the
given hypothesis (ii) and thus (ii) must be true.
(iii)  (iv)

Let (iii) be true.
Suppose X = A B

where A  , B  , A B =  = A B.
Then clearly X = A B

where A and B are non-empty closed sets.

Also A B = 

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