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I just started with Schaum's Outline of General Topology by Seymour Lipschutz a few days ago, and now I am working with chapter 2 about functions.

Most of the material I have already seen in other books, but some definitions are new to me. One such definition is the definition of the direct product (p.19):

The Cartesian product of an indexed class of sets $\mathcal{A} = \{A_i \mid i \in I\}$, denoted by $\prod_{i\in I} A_i$ is the set of all functions $p: I \rightarrow \cup_i A_i $ s.t. $p(i) = a_i$. We denote such an element of the Cartesian product $p = <a_i \mid i \in I>$.

After meditating on this for a while, I realized that it is just a generalization of the "easier" Cartesian product which was defined using ordered pairs (is this correct?). Since we do not assume anything about the index set, it can presumably be anything, even uncountable, I take it (is this correct?). It also seems that this definition is equivalent to the one with ordered pairs if the index set consists of natural numbers.

It looked a little scary at first, but I can get used to thinking about the direct product in terms of functions (I think); so far so good. What has puzzled me (even after meditating on the issue) is the following (also p.19):

For each $i_0 \in I$ there exists a function $\pi_{i_0}$, called the $i_0$th projection function, from the set $\prod_{i\in I} A_i$ into the $i_0$th coordinate set $A_{i_0}$ defined by $\pi_{i_0}(<a_i \mid i \in I>)=a_{i_0}$.

What puzzles me is the purpose of this projection function. It seems to take a function (the $p$ defined above, i.e., a tuple) and produce a coordinate; that is $\pi_{i_0}(p) = a_{i_0} = p(i_0)$ if I understood correctly.

So if we can already recover the coordinate $a_{i_0}$ from information contained in $p$ and $i_0$, why do we need $\pi_{i_0}$? In what situation would we use it? Also, are we always guaranteed of the existence of $\pi_{i_0}$?

I would be grateful if you would share your expertise with me in understanding this definition!

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    $\begingroup$ The example I like to use is the index set $\{x,y,z\}$ for the usual $x,$ $y$, and $z$ coordinates we associate with $\mathbb{R}^3$. Each function maps the coordinates to the point. There is one function for each point. $\endgroup$ Aug 24 '20 at 21:53
  • $\begingroup$ @CyclotomicField Thank you for the reaction! You're right that I should try and use something that is familiar as concrete example. So if $I=\{x,y,z\}$ then the Cartesian product by this definition would be $\mathbb{R}^3 = \prod_{i\in I} {\mathbb{R}_i}$ which is the set of all functions $p$ s.t. $p(x) = a_x \in \mathbb{R}_x, p(y) = a_y \in \mathbb{R}_y, p(z) = a_z \in \mathbb{R}_z$. Thus as you say, there is one function per point (the notation got a little weird, but I hope it is clear). $\endgroup$ Aug 24 '20 at 22:18
  • $\begingroup$ Then we have three projection maps, i.e., functions $\pi_x, \pi_y, \pi_z$ that map the points (the functions $p$) to their coordinates, right? I think this is what I didn't realize before. $\endgroup$ Aug 24 '20 at 22:19
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    $\begingroup$ The projection maps are the function restricted to a singleton domain, ie the coordinate you're projecting onto. The projection differs from the index function in domain only but it's still a slightly different function than $p$. $\endgroup$ Aug 24 '20 at 22:26
  • $\begingroup$ Thank you for the clarification, I really appreciate it! $\endgroup$ Aug 24 '20 at 22:29
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You're right that this general definition (the Cartesian product as a special set of functions) generalises the "easier" product based on ordered pairs: if we take as the domain $I$ a two point set (doubleton) like $I=\{0,1\}$ the general definition talks about functions $f: \{0,1\} \to A_0 \cup A_1$ with $f(0)\in A_0$ and $f(1) \in A_1$, which we can "encode" or summarize as an ordered pair $I(f):=(f(0), f(1)) \in A_0 \times A_1$, which also uniquely gives two points, one from $A_0$ and the other from $A_1$. This $f \to I(f) \in A_1 \times A_2$ is clearly a bijection between the general product $\prod_{i \in \{0,1\}} A_i$ and $A_1 \times A_2$ (the two values uniquely give the components of the pair, and the pair gives us a unique way to define a function on $\{0,1\}$ etc.). Projections is a term we know for ordered pairs: for the set $X \times Y$ the projection onto $X$, say $\pi_X$ is the function $\pi_X: X \times Y \to X$ that maps $(x,y)$ to its "$X$-component", namely $x$. Similarly, $\pi_Y: X \times Y \to Y$ is defined by $\pi_Y((x,y))= y$ for all pairs.

So the values of the projections uniquely determine the pair, and this is often expressed as a so-called universal property of the projections $\pi_X,\pi_Y$ for the product $X \times Y$:

If $f_1: Z \to X$ and $f_2: Z \to Y$ are two arbitrary functions with common domain $Z$, there is a unique function (denoted) $f_1 \times f_2: Z \to X \times Y$ such that $$\pi_X \circ (f_1 \times f_2) = f_1 \text{ and } \pi_X \circ (f_1 \times f_2) = f_2\tag{1}$$

And this property is the "essential" defining property of $X \times Y$ in the sense that if we have any set $U$ with two maps $\pi_X : U \to X$ and $\pi:U \to Z$ that also obeys the universal property as before, then there is a natural bijection between $U$ and $X \times Y$ that "respects the projections". If you want to know more about such things, category theory is the place to be, and the product of $X \times Y$ is called a limit there. For our set $U=\prod_{i \in \{0,1\}} A_i$ for instance, we use $\pi_{A_0} = f(0)$ and $\pi_{A_1} = f(1)$ and this works and has the universal property, so the set of functions "essentially is" (is naturally isomorphic to one would say in category theory) to $A_1 \times A_2$.

The reason I mention all this is to illustrate the essential rôle of the projection maps: they essentially determine how a product behaves wrt functions we want to define into it. We generalise this property to arbitrary index sets and the universal property becomes (we define, as in your text $\pi_i(f) = f(i) \in A_i$ as a set of functions from the product to $A_i$ resp.):

If for all $i \in I$ we have a function $f_i: Z \to A_i$ with common domain $Z$, there is a unique function $h: Z \to \prod_{i \in I} A_i$ such that:

$$\forall i \in I: \pi_i \circ h = f_i$$

So the projections are the analogues of the maps $(x,y) \to x$ and $(x,y) \to y$ we know from standard "pair"-products and they have the same abstract property.

Back to topology: it will also turn out that the projection maps will determine the topology on the product based on topologies on the $A_i$ with a universal property that is essentially the same.

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