Open Sets in the Product Toplogy

I have a problem that is two fold. First, I am not comfortable with the notation that my textbook uses. I am familiar with another notation of product space (from say Munkres which I read on my own first). Secondly, I am not really comfortable wit the product toplogy. I can get away most of the time, i.e., when the products are finite and hence equal to the box topology but when I must use the product topology I get a headache :-) So I am hoping that maybe somebody here could help me.

First off here, is the begining of the problem I am trying to solve:

Let $\displaystyle \mathcal{K}$ be a set of continuous functions from $\displaystyle (X, \tau)$ into $\displaystyle [0,1]$.

Define $\displaystyle K : X \to [0,1]^\mathcal{K}$ by $\displaystyle K(x)(f) = f(x)$.

I then translated it into what I think is right in notation that I am familiar with. Here it is:

Consider $\displaystyle \{ f_\alpha \}_{\alpha \in J}$, an indexed set of continuous functions $\displaystyle f_\alpha : X \to [0,1]$ and define $\displaystyle f : X \to [0,1]^J$ by $\displaystyle f(x) = (f_\alpha(x))_{\alpha \in J}$.

Assuming that is right, my next problem is to show that $\displaystyle K$ or in my notation $\displaystyle f$ is continuous. But what are the open sets in $\displaystyle [0,1]^J$. I think they are $\displaystyle \pi^{-1}_\alpha(U)$ where $\displaystyle U$ is open in $\displaystyle [0,1]$? Are these open sets functions? I understand that the "points" in $\displaystyle [0,1]^J$ are functions (i.e. the $\displaystyle f_\alpha$'s), but what are the open sets. Are they an uncountable number of continuous functions. Let me stop here before I start sounding really dumb.

Thanks for the help.