Continuous Functions and Topologies

• September 3rd 2013, 11:52 AM
Aryth
Continuous Functions and Topologies
We have a definition for continuity in terms of open sets of topologies, but I had a question about it. Here's the definition:

Let $(X,\Omega)$ and $(Y,\Theta)$ be topological spaces. A function $f: X \to Y$ is continuous relative to $\Omega$ and $\Theta$ provided $f^{-1}(U) \in \Omega$ for every $U \in \Theta$.

My question is this: Is $f^{-1} : Y \to X$ continuous relative to $\Theta$ and $\Omega$ provided $U \in \Theta$ for every $f^{-1}(U) \in \Omega$? Or is there something else you have to do to that sentence to make it true?
• September 3rd 2013, 12:20 PM
Plato
Re: Continuous Functions and Topologies
Quote:

Originally Posted by Aryth
We have a definition for continuity in terms of open sets of topologies, but I had a question about it. Here's the definition:
Let $(X,\Omega)$ and $(Y,\Theta)$ be topological spaces. A function $f: X \to Y$ is continuous relative to $\Omega$ and $\Theta$ provided $f^{-1}(U) \in \Omega$ for every $U \in \Theta$.

My question is this: Is $f^{-1} : Y \to X$ continuous relative to $\Theta$ and $\Omega$ provided $U \in \Theta$ for every $f^{-1}(U) \in \Omega$? Or is there something else you have to do to that sentence to make it true?

This is a case in which the function notation is getting into the way of understanding.

You have a function $f:X\to Y$.
But then $f^{-1}$ maps ${P}(Y)\to {P}(X)$, i.e. between power sets.

Continuity is defined on topological spaces. What is the topology on ${P}(Y)~?$
• September 3rd 2013, 12:59 PM
HallsofIvy
Re: Continuous Functions and Topologies
The very first time I was asked to present a proof in a Graduate school class, it was precisely a problem involving $f^{-1}(A)$ where A was in the image of f. I did the whole problem assuming that f was invertible (they said " $f^{-1}$" didn't they?)!