# A couple questions..

• Sep 4th 2008, 03:20 PM
Proof_of_life
A couple questions..
I am taking an undergraduate algebraic geometry course this semester, and I will probably be on here a lot! Anyways, my first question is if this is the correct forum to be posting my future questions on? If not, if someone could move my post, I would greatly appreciate it!

I have a couple small examples that I need to show for next class but I am a little rusty on some definitions and how to interpret them into my solutions.

(1) Find a continuous bijection which is not a homomorphism.

(2) Show that the open ball $\displaystyle B^n =${$\displaystyle (x_1,....,x_n) \in \mathbb{R}^n | x^{2}_1 + ... + x^{2}_n < 1$} in $\displaystyle \mathbb{R}^n$ is homomorphic to $\displaystyle \mathbb{R}^n$

I am so lost it's not even funny, especially on the 2nd problem...
Thanks for looking.
• Sep 4th 2008, 03:59 PM
Plato
Let us start out in a very basic mode.
Can you define what a "homomorphism" is?
Can you give theorems related to that definition?
• Sep 4th 2008, 06:18 PM
Proof_of_life
We say a continuous map $\displaystyle f: X \to Y$ is a homomorphism if $\displaystyle f$ is 1-1 and onto and $\displaystyle f^{-1}: Y \to X$ is continuous. We use the notation $\displaystyle X\approx Y$ to denote homomorphism.
• Sep 8th 2008, 02:37 AM
HallsofIvy
Quote:

Originally Posted by Proof_of_life
We say a continuous map $\displaystyle f: X \to Y$ is a homomorphism if $\displaystyle f$ is 1-1 and onto and $\displaystyle f^{-1}: Y \to X$ is continuous. We use the notation $\displaystyle X\approx Y$ to denote homomorphism.

That's a homEomorphism. In other words, you are looking for a one-to-one, onto continuous function whose inverse is not continuous.