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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.

nmasanta
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  • Possible duplicate of Meaning of a set in the exponent – Rushabh Mehta Jul 30 '19 at 01:02
  • @nmasanta: Was there an actual point to your edit, other than working towards several badges and removing this question from the review queue? – Asaf Karagila Jul 30 '19 at 06:30
  • @Asaf Karagila: The actual point of my edit is to prevent this question from being closed. As I have sufficient badges so I think, I have the right to express my thoughts. Thank you. – nmasanta Jul 30 '19 at 06:55
  • @nmasanta: You're preventing nothing, you're just pushing it off the review queue. It can still be closed. And while you have the right to express your thoughts, others have too. And by pushing it off the review queue like that you're effectively saying that your thoughts are more important than others. – Asaf Karagila Jul 30 '19 at 06:56
  • @Asaf Karagila: I had no intention to do so. If it happens, then I am sorry about that. – nmasanta Jul 30 '19 at 07:03

1 Answers1

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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.

Frank
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