# Topology

Proof that (a,b) is not homeomorphic to [a,b]. Assume that they are. Thus we have a continuous, continuously invertible function f from [a,b] to (a,b). Now because f is invertible it is bijective thus f(a)=f(b) is false. Let f(a)<f(b) so that a<f(a)<f(b)<b. Not for f to be invertible we must have a c in (a,b) such that f(c) is in (f(b),b) so that f(b)<f(c). Now by the Intermediate Value Theorem there is a d in (a,c) such that f(d)=f(b). Thus f fails to be invertible if f(a)<f(b).

Note by Samuel Queen
5 years ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

I like this proof. Usually an invariant like compactness is used, but this is more Analysis based.

- 3 years, 4 months ago

The same result holds if f(a)>f(b).

- 5 years ago