13.9. CONNECTED SETS 247

13.9 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. This iscalled separated. Connected sets are defined in terms of not being separated. This is whytheorems about connected sets sometimes seem a little tricky.

Definition 13.9.1 Let A be a nonempty subset Rn. Then A is defined to be the intersectionof all closed sets which contain A. This is called the closure of A. Note the whole space, Rn

is one such closed set which contains A.

Lemma 13.9.2 Let A be a nonempty set in Rn. Then A is a closed set and

A = A∪A′

where A′ denotes the set of limit points of A.

Proof: First of all, denote by C the set of closed sets which contain A. Then

A = ∩C

and this will be closed if its complement is open. However,

AC= ∪

{HC : H ∈ C

}.

Each HC is open and so the union of all these open sets must also be open. This is becauseif x is in this union, then it is in at least one of them. Hence it is an interior point of thatone. But this implies it is an interior point of the union of them all which is an even largerset. Thus A is closed.

The interesting part is the next claim. First note that from the definition, A ⊆ A so ifx ∈ A, then x ∈ A. Now consider y ∈ A′ but y /∈ A. If y /∈ A, a closed set, then there existsB(y,r)⊆ AC

. Thus y cannot be a limit point of A, a contradiction. Therefore,

A∪A′ ⊆ A

Next suppose x∈A and suppose x /∈A. Then if B(x,r) contains no points of A differentthan x, since x itself is not in A, it would follow that B(x,r)∩A = /0 and so recalling thatopen balls are open, B(x,r)C is a closed set containing A so from the definition, it alsocontains A which is contrary to the assertion that x ∈ A. Hence if x /∈ A, then x ∈ A′ andso

A∪A′ ⊇ A ■

Now is a definition about what it means to not be connected. This is called separated.

Definition 13.9.3 A set, S in Rn, 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.