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

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

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

My question is this: Is $\displaystyle f^{-1} : Y \to X$ continuous relative to $\displaystyle \Theta$ and $\displaystyle \Omega$ provided $\displaystyle U \in \Theta$ for every $\displaystyle f^{-1}(U) \in \Omega$? 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 $\displaystyle f:X\to Y$.

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

Continuity is defined on topological spaces. What is the topology on $\displaystyle {P}(Y)~?$

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 $\displaystyle f^{-1}(A)$ where A was in the image of f. I did the whole problem **assuming** that f was invertible (they said "$\displaystyle f^{-1}$" didn't they?)!