Isn’t Halmos wrong here?

Question

In his Naive Set Theory, under Section 9, Families, he states the following:

Suppose that $\{ X_i\}$ is a family of sets $(i\in I)$ and let $X$ be its Cartesian product. If $J$ is a subset of $I$, then to each element [$x$] of $X$ there corresponds ... [an] element, say $y$, of $\prod_{i\in J} X_i$ is obtained by simply restricting that function [$x$] to $J$. ... The correspondence $x\to y$ is called the projection from $X$ onto $\prod_{i\in J} X_i$; we shall temporarily denote it by $f_{J}$. If, in particular, $J$ is a singleton, say $J=\{ j\}$, then we shall write $f_{j}$ ... for $f_{J}$. ... if $x\in X$, $\bbox[aqua]{\text{the value of } f_{j}\text{ at } x,\text{that is } x_j}$, is also called ...

Isn’t the highlighted portion wrong?

Should it not read “$\bbox[yellow]{\ldots\text{the value of } f_j\text{ at } x,\text{that is } \{(j,x_j)\}\ldots}$ ”?

Answer 1

Your quote doesn’t exactly match what I see in Halmos’s book here, but the part you highlight looks fine to me.

I think where your question arises is in being precise about how to denote elements of a Cartesian product, which Halmos addresses on page $36$. Note in particular where he says

Explicitly, we write $y_i=x_i$ whenever $i\in J$.

So he is saying, I think, that when $y$ is an element of the Cartesian product (and $y$ is thus a function from the index set to the union of the product’s factors), then if the value of (the function) $y$ on the index $i$ is $x_i$, we denote the restriction of the function $y$ to a singleton domain $\{i\}$ as $x_i$, even though it is technically not a value in $X_i$ itself but is a function with domain $\{i\}$ that has value $x_i$ on its unique domain element. This is not a problem because there is a one-to-one correspondence between values of $X_i$ and functions from $\{i\}$ to $X_i$.

For example, if $X_1=X_2=X_3=\mathbb R$, the function $f_2$ would project an element of ${\mathbb R}^3$ onto the coordinate corresponding to the copy of $\mathbb R$ called $X_2$. Modern mathematicians typically write an element $x$ of ${\mathbb R}^3$ in coordinate notation as $(x_1,x_2,x_3)$, so with that convention, $f_2((x_1,x_2,x_3))$ (usually simplified to $f_2(x_1,x_2,x_3)$ would be $x_2$, even though using Halmos’s generalization of Cartesian product, $f_2(x)$ would be a function from $\{2\}$ to $\mathbb R$ with value $x_2$ on the only possible input $2$. I think you want to write this function as $\{(2,x_2)\}$, but that’s cumbersome and Halmos explains that he will write that as $x_2$.