70 CHAPTER 2. BASIC TOPOLOGY AND ALGEBRA

2.10 Connected SetsStated informally, connected sets are those which are in one piece. In order to define whatis meant by this, I will first consider what it means for a set to not be in one piece. Thisis called separated. Connected sets are defined in terms of not being separated. This iswhy theorems about connected sets sometimes seem a little tricky. It is defined in terms ofwhat it is not, rather than what it is. Much of this works fine in more general settings, butI will only consider the context of Rp because this is what is of interest in this book and Idon’t want to keep changing the context in order to get the most general versions. Now isa definition about what it means to not be connected. This is called separated.

Definition 2.10.1 A set, S in Rp, is separated if there exist sets A,B such that

S = A∪B, A,B ̸= /0, and A∩B = B∩A = /0.

In this case, the sets A and B are said to separate S. A set is connected if it is not separated.Remember A denotes the closure of the set A.

One of the most important theorems about connected sets is the following.

Theorem 2.10.2 Suppose U is a set of connected sets and that there exists a pointp which is in all of these connected sets. Then K ≡ ∪U is connected.

Proof: The argument is dependent on Lemma 2.2.15. Suppose

K = A∪B

where Ā∩B = B̄∩A = /0,A ̸= /0,B ̸= /0. Then p is in one of these sets. Say p ∈ A. Then ifU ∈U , it must be the case that U ⊆ A since if not, you would have

U = (A∩U)∪ (B∩U)

and the limit points of A∩U cannot be in B hence not in B∩U while the limit points ofB∩U cannot be in A hence not in A∩U . Thus B = /0. It follows that K cannot be separatedand so it is connected. ■

The intersection of connected sets is not necessarily connected as is shown by the fol-lowing picture.

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Theorem 2.10.3 Let f : X→Rm be continuous where X is connected. Then f(X) isalso connected.