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 and be topological spaces. A function is continuous relative to and provided for every .

My question is this: Is continuous relative to and provided for every ? 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

and

be topological spaces. A function

is continuous relative to

and

provided

for every

.

My question is this: Is

continuous relative to

and

provided

for every

? 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 .

But then maps , i.e. between power sets.

Continuity is defined on topological spaces. What is the topology on

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 where A was in the image of f. I did the whole problem **assuming** that f was invertible (they said " " didn't they?)!