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