## The Topology of (is Weird)

Definition 16.2.10 (Connected)   Let be a topological space. A subset  of  is if there exist open subsets with and with and nonempty. If  is not disconnected it is .

The topology on is induced by , so every open set is a union of open balls

Recall Proposition 16.2.8, which asserts that for all ,

This translates into the following shocking and bizarre lemma:

Lemma 16.2.11   Suppose and . If and , then .

Proof. Suppose and . Then

You should draw a picture to illustrates Lemma 16.2.11.

Lemma 16.2.12   The open ball is also closed.

Proof. Suppose . Then so

Thus the complement of is a union of open balls.

The lemmas imply that is , in the following sense.

Proposition 16.2.13   The only connected subsets of are the singleton sets for and the empty set.

Proof. Suppose is a nonempty connected set and are distinct elements of . Let . Let and be the complement of , which is open by Lemma 16.2.12. Then and satisfies the conditions of Definition 16.2.10, so  is not connected, a contradiction.

William Stein 2004-05-06