Showing that a given function is continous over a certain topology

[trap101]

Consider the maps h: R^w (omega) ---> R^w (omega) , h (x1, x2, x3,...) = (x1,4x2, 9x3,...)
g: (same dimension mapping) , g (t) = (t, t, t, t, t,...)

Is h continuous whn given the product topology, box topology, uniform topology?

For the life of me i am having trouble trying to understand what i need to do to accomplish this. I know that the definition of continuous in topology is that the pre image f^-1 (v) of an open set v has to be open, but how do i apply that to this sort of situation?

Take the function h, am i assuming that (x1, 4x2, 9x3,...) is my open set or am i going to take an arbitrary basis element of my open set that looks like (x1, 4x2, 9x3,...) apply the preimage to that, and then if it is open then i can conclude the function is continuous?

I saw a solution for the function g, and they said it is not open in the box topology. They took th open set [-1/(n^2) , 1/(n^2)] and said taking the inverse image of g the set is {0}. How did they do this? I fail to see the connection. Please help I've been stuck here all day tryimg to make sense of this.

 

[RUber]

I imagine that for g, they were taking the limit as $n \to \infty$. This would give you an open set for the image, but in the limit, the pre-image is the closed singleton {0}.

 

[trap101]

I imagine that for g, they were taking the limit as $n \to \infty$. This would give you an open set for the image, but in the limit, the pre-image is the closed singleton {0}.

Yes, but how do they get that singleton is my question. Is it that you look at each inverse image of each coordinate function individually and take the intersection of those , which produces the common point 0?

 

[RUber]

Assume that $U$ is an open set $U=(-1/n^2, 1/n^2)$, for $V=g^{-1}(U)$ to be open, there must be $\epsilon \in V, \, \epsilon \neq 0$, such that $g(\epsilon) \in U$. But for any epsilon, there is an $n$ that makes $g(\epsilon) \notin U$.
You could also think of $U = (-\epsilon, \epsilon)$ if this helps.
Wikipedia uses exactly this example to describe how box topology fails at continuity tests: http://en.wikipedia.org/wiki/Box_topology