[Math] What does it mean when a function $f$ has a subscript that is an indexing set $A$? That is, $f_A$.

functionsnotation

I'm reading Intro to Topology by Mendelson.

I'm having trouble understanding certain notation he uses for a particular problem. To put it into context, here is the problem at hand

Let $\{X_\alpha\}_{\alpha\in A}$ be an indexed family of topological spaces and set $X=\prod_{\alpha\in A} X_\alpha.$ For each $\alpha\in A$ let $f_\alpha:I\to X_\alpha$ be a path in $X_\alpha$. Set $(f_A(t))(\alpha)=f_\alpha(t)$ so that $f_A:I\to X$. Show that $f_A$ is a path in $X$.

I don't necessarily need any help with this problem yet, because I haven't even tried it. I looked through the book for that type of notation, but was unsuccessful. Could someone please point out what the definition for $f_A$ is? Is it specific to this problem or a general type of notation?

Best Answer

The definition of $f_A$ is right in the text you quoted:

Set $(f_A(t))(\alpha)=f_\alpha(t)$ so that $f_A: I\to X$.

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