What does $X^Y$ mean?

Question

I have been reading about the Stone-Čech compactification recently and one way of constructing it is by considering the map $~h:X\rightarrow I^{C}:x\mapsto(fx)_{f\in C}~$ where $~I~$ is the closed unit interval and $~C~$ the set of continuous maps from $~X~$ to $~I~$ and defining $~\beta X=\overline{hX}~$.

My problem is that I do not fully understand what $I^{C}$ is.

Right now, my understanding is that it is the set of all maps from $C$ to $I$ but I am not really convinced.

Especially since the map $h$ maps $x$ to $(fx)_{f\in C}$ which makes me think that the codomain of $h$ is $I$ rather than $I^C$.

And more generally what does $X^Y$ mean? ($X,Y$ are topological spaces).

Thank you in advance.

Answer 1

It means the set of functions from $Y$ to $X$. So your suspicion was correct.

EDIT $ \ \ $The notation $(f(x))_{f \in C}$ is a shorthand for the map $E: C \rightarrow I$ defined by $E(f):= f(x)$, called the evaluation map.