##

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:

*Proof*.
Suppose

and

. Then

a contradiction.

You should draw a picture to illustrates Lemma 14.2.11.

*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