Classnotes for October 13

From 0506Topology

#	Week of...	Links (edit)
Fall
1	Sep 12	About, Tue, Thu, Std2Disc
2	Sep 19	Tue, Thu, HW1, 14 Sets
3	Sep 26	Tue, Thu, Photo
4	Oct 3	Tue, Thu
5	Oct 10	HW2, Tue, Thu
6	Oct 17	Tue, Thu, HW3
7	Oct 24	Mon, Tue, Thu
8	Oct 31	Tue, Thu, HW4
9	Nov 7	TE1, Tue, Thu
10	Nov 14	Tue, Thu, HW5
11	Nov 21	Tue, Thu
12	Nov 28	Tue, Thu, HW6
13	Dec 5	Tue, Thu
E	Dec 12	TE2
Spring
14	Jan 9	Tue, IT83, Thu, HW7
15	Jan 16	Tue, Thu
16	Jan 23	Tue, HW8, Thu
17	Jan 30	Tue, Thu
18	Feb 6	TE3, Tue, Thu
19	Feb 13	Tue, Thu
R	Feb 20
20	Feb 27	Tue, Thu, HW9
21	Mar 6	Tue, Thu, HW10
22	Mar 13	Tue, Thu
23	Mar 20	Tue, Thu, HW11
24	Mar 27	Tue, Thu
25	Apr 3	Tue, Thu, HW12
26	Apr 10	Tue, Thu
Study	Apr 17	Office Hours
Exams	Apr 24	Final, PM

Table of contents

1 Basic Properties of Connected Sets

2 Products of Connected Sets

3 Scanned Notes

3.1 JPG

[edit]

Basic Properties of Connected Sets

We recall, from last class, the definition of a connected set and a simple example:

Definition: A set is connected if it is not a non-trivial disjoint union of open sets.

Theorem: $[0,1]$ is connected in $\mathbb{R}$ .

As one might expect, continuous maps preserve connectedness.

Theorem: If $f\colon X \rightarrow Y$ is continuous, then $X$ is connected implies that $f (X)$ is connected.

Proof: If $f (X) =$ $A \cup B$ with $A$ and $B$ disjoint and open, $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$ with $f - 1 (A)$ and $f - 1 (B)$ open and disjoint. That is, $X$ is not connected with is a contradiction.

Intermediate Value Theorem: If $f\colon [a,b] \rightarrow \mathbb{R}$ where $f (a) < 0 < f (b)$ , then $\exists \; x \in [a,b]$ such that $f (x) = 0$ .