1. ## [SOLVED] Continuous Functions/Maps

I need help with what I am sure just a simple tweek in concept can fix.

In topology, we define a continuous map as follows:

Definition: Let $X$ and $Y$ be topological spaces. A map $f: X \to Y$ is called continuous if the inverse image of open sets is always open. (says the book, "Topology" by Klaus Janich)

That is, (and this is not in the book) $f: (X_1, Y_1) \to (X_2,Y_2)$ is a continuous map if for every set $U \in Y_2$ we have $f^{-1}(U) \in Y_1$

My question is, how can we prove that this definition of continuity is equivalent to the calculus $\epsilon - \delta$ definition of continuity?

I'm pretty sure I have to say something about "metric spaces" and "neighborhoods" but there is a small conceptual leap i can't make to connect it all.

If necessary, I can supply any definition I am working with upon request, but I don't want someone to write the proof for me, but in layman's terms, tell me how I should think about it, what conceptual links should I be making between calculus and topology in this respect?

2. Originally Posted by Jhevon
I need help with what I am sure just a simple tweek in concept can fix.

In topology, we define a continuous map as follows:

Definition: Let $X$ and $Y$ be topological spaces. A map $f: X \to Y$ is called continuous if the inverse image of open sets is always open. (says the book, "Topology" by Klaus Janich)

That is, (and this is not in the book) $f: (X_1, Y_1) \to (X_2,Y_2)$ is a continuous map if for every set $U \in Y_2$ we have $f^{-1}(U) \in Y_1$

My question is, how can we prove that this definition of continuity is equivalent to the calculus $\epsilon - \delta$ definition of continuity?

I'm pretty sure I have to say something about "metric spaces" and "neighborhoods" but there is a small conceptual leap i can't make to connect it all.

If necessary, I can supply any definition I am working with upon request, but I don't want someone to write the proof for me, but in layman's terms, tell me how I should think about it, what conceptual links should I be making between calculus and topology in this respect?
In a metric space a set $S$ is open if for every point in the set there is an open ball centred on the point which is a subset of $S$.

The $\epsilon - \delta$ definition of continuity is a definition in terms of open balls (which are themselves open sets). But an arbitary union of open sets is open, which should be sufficient to complete a proof that the $\epsilon - \delta$ definition of continuity is equivalent to the topological defintion for the usual topology on $\mathbb{R}$.

RonL

Note: the $\epsilon - \delta$ definition of continuity is something like: The inverse image of an open ball contains an open ball.

3. Originally Posted by CaptainBlack
In a metric space a set $S$ is open if for every point in the set there is an open ball centred on the point which is a subset of $S$.

The $\epsilon - \delta$ definition of continuity is a definition in terms of open balls (which are themselves open sets). But an arbitary union of open sets is open, which should be sufficient to complete a proof that the $\epsilon - \delta$ definition of continuity is equivalent to the topological defintion for the usual topology on $\mathbb{R}$.

RonL
i believe i see what you are saying, and i kind of had similar thoughts, but somehow i don't see what inverse mappings have to do with this. that's the link i was looking for. i want to explicitly tie the relationship between a continuous function and it's inverse into all this, and make similar ties with topological spaces

4. Originally Posted by Jhevon
i believe i see what you are saying, and i kind of had similar thoughts, but somehow i don't see what inverse mappings have to do with this. that's the link i was looking for. i want to explicitly tie the relationship between a continuous function and it's inverse into all this, and make similar ties with topological spaces
The $\epsilon - \delta$ definition may be rephrased as the inverse image of an open ball centred on $f(x_0)$ contains an open ball centred on $x_0$. Then the arbitary union of open sets property gives the result that an $\epsilon - \delta$ continuous function is a continuous function wrt the usual topology on the reals.

RonL