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 be a set of continuous functions from into .

Define by .

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

Consider , an indexed set of continuous functions and define by .

Assuming that is right, my next problem is to show that or in my notation is continuous. But what are the open sets in . I think they are where is open in ? Are these open sets functions? I understand that the "points" in are functions (i.e. the '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.