## The Topology of (is Weird)

Definition 14.2.10 (Connected)   Let be a topological space. A subset of is disconnected if there exist open subsets with and with and nonempty. If is not disconnected it is connected.

The topology on is induced by , so every open set is a union of open balls Recall Proposition 14.2.8, which asserts that for all , This translates into the following shocking and bizarre lemma:

Lemma 14.2.11   Suppose and . If and , then .

Proof. Suppose and . Then a contradiction. You should draw a picture to illustrates Lemma 14.2.11.

Lemma 14.2.12   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 totally disconnected, in the following sense.

Proposition 14.2.13   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 14.2.12. Then and satisfies the conditions of Definition 14.2.10, so is not connected, a contradiction. William Stein 2012-09-24