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The set of all homomorphisms between two spaces $X$ and $Y$ is denoted as $\text{Hom}(X,Y)$, the set of endomorphisms of a space $X$ is denoted as $\text{End}(X)$,... these are standard notations, but what about the set of all maps between two spaces $X$ and $Y$ (without any extra structure) Are $\text{Map}(X,Y)$ or $\text{Fun}(X,Y)$ standard notations?

This might be an stupid question, but I want to make sure I am using the right notation, or at least the most usual notation for this spaces. Thanks for your help.

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    $\begingroup$ Notations are defined. If you feel weird when using these notations, please define them first. I I have no impression about a notation for the function space between $X$ and $Y$. $\endgroup$
    – xfireskyx
    Jan 24, 2022 at 16:59
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    $\begingroup$ you can use $Y^X$ $\endgroup$
    – janmarqz
    Jan 24, 2022 at 17:01
  • $\begingroup$ @xfireskyx Yeah, I know, but when possible it is better to use standard notations. For example, I could define the derivative of $f$ as $\hat{f}$, but it is not standard and no one will understand it. $\endgroup$
    – Marcos
    Jan 24, 2022 at 17:01
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    $\begingroup$ It all depends on context. Those are certainly used and are more common to see on the algebra side of things. From a combinatorics side of things you might see $Y^X$ used to denote the set of functions from $X$ to $Y$ instead which assists with counting as one would have $|Y^X|=|Y|^{|X|}$ $\endgroup$
    – JMoravitz
    Jan 24, 2022 at 17:01
  • $\begingroup$ @JMoravitz Right, thanks for your answer. $\endgroup$
    – Marcos
    Jan 24, 2022 at 17:06

2 Answers 2

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Common ways to denote the set of all functions from a set $X$ to a set $Y$ that I have encountered (some of which have already been mentioned) are $\mathcal{F}(X;Y)$, $\text{Map}(X;Y)$ and $Y^X$. Then you can define $\mathcal{F}(X):=\mathcal{F}(X;X)$ etc. I don't think there is a 'standard' way of presenting such functions that has anywhere near a consensus. I would personally use whichever notation feels most natural, depending on the context.

The $Y^X$ notation comes from the fact that you are choosing one value of $Y$ for each value of $X$, so you can kind of think of it is an '$X$-fold Cartesian product' in this sense i.e. $X$ copies of the set $Y$.

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    $\begingroup$ Thanks, that was what I needed. $\endgroup$
    – Marcos
    Jan 24, 2022 at 17:15
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As mentioned in the comments, the notation $Y^X$ is standard for the set of maps from $X$ to $Y$. Moreover, there is no other common notation for this.

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