Understanding quotient topology and product topology (in the infinite case) I have troubles understanding the concepts of quotient topology and product topology (in the infinite case). 
I know that we want to give a topology to new spaces built from the old ones, but the thing is that I can't figure out why is the definition for quotient topology natural since we only require that the canonical projection should be continuos (I think this definition is given tersley), and on the other hand I don't understand why does the box topology doesn't work in the infinite case so we have to define a very special topology where you say that you have infinite tuples where most of them are the space itself so, How could this work?
And can you recommend some exercises to put in practice this concepts please.
Thanks a lot in advance.
 A: A guiding principle in modern mathematics is that studying morphisms between objects is even more important than studying the objects themselves. In the category of topological spaces, this means that one should be more interested in continuous maps between spaces than in the spaces themselves. From this point of view, it is natural to endow certain spaces with a topology that is the "least trivial possible" and that makes canonical maps from/to this space continuous.
The quotient topology is an example of that. Of course, we could endow $X/{\sim}$ with the trivial (indiscrete) topology, and then the canonical map $X \to X/{\sim}$ would be continuous, but this is of little interest as in fact any map $Y \to X/{\sim}$ would be continuous. In a way we want to give the quotient space the most optimal topology that makes the canonical projection continuous. In this case "most optimal topology" would mean "finest topology". One should also be content with the definition of this topology because in practice it truly reflects how we like to think of quotient spaces as being spaces glued together under an equivalence relation: gluing together two ends of a line segment yields a circle, gluing the boundary a disk to a point yields a sphere, etc.
Similarly, and especially after considering how product objects generally behave in other categories, one would want to endow $\Pi_{i \in I} X_i$ with the "most optimal" topology that makes the projection maps $\Pi_{i \in I} X_i \to X_j$ continuous. This topology is the product topology. The box topology, on the other hand, can be deceiving in the infinite case because one can have a discontinuous map $Y \to \Pi_{i \in I}X_i$ such that all its component functions $Y \to X_j$ are continuous. These are the kinds of things we want to avoid.
A: Both of these can be understood via universal properties. Let's look at the product first.
If you want to form the product of two sets $S$ and $T$, what do you do? You form the set $S\times T = \{(s,t)\mid s\in S, t\in T\}$. However, rather than just look at the set $S\times T$ itself, one should think about the properties that $S\times T$ satisfies. You can convince yourself that it in fact satisfies the following:

Let $X$ be a set, and let $f : X\to S$ and $g : X\to T$ be two maps of sets. Then there exists a unique map of sets $h : X\to S\times T$ such that $f = \pi_S\circ h$ and $g = \pi_T\circ h$, where $\pi_S : S\times T\to S$ is the natural projection sending $(s,t)$ to $s$ (similarly for $\pi_T$).

This can be generalized to create the notion of a product of arbitrarily many sets $\{S_i\}_{i\in I}$: you ask for the product $\prod_{i\in I} S_i$ to be a set such that for any set $X$ with maps $f_i : X\to S_i$ for each $i$, you have a unique map $X\to\prod_{i\in I} S_i$ such that factorizations of each $f_i$ analogous to the above factorization hold.
If we categorify, we can replace "set" by "object" and "map of sets" by "morphism" in any category, and get the notion of product in an arbitrary category:

An object $X$ in a category is the product of a family $\{X_i\}_{i\in I}$ of objects if and only if there exist morphisms $\pi_i : X \to X_i$ for all $i$, such that for every object $Y$ equipped with morphisms $f_i : Y \to X_i$ for all $i$ there exists a unique morphism $f : Y \to X$ such that $f_i = \pi_i\circ f $ for all $i\in I$. 

So, replacing "set" by "topological space" and "map of sets" by "continuous map," we obtain the [categorical] definition of the product of topological spaces. You can verify that in fact, the explicit description of the product topology that you know satisfies the universal property I have described, and so deserves to be called the product of topological spaces.
Note that the box topology does exist on an infinite product of spaces (that is, the box topology is a topology on the product of spaces considered as a set), but it does not satisfy the universal property described above. In particular, there are "too many" open sets. (As you may have heard, the box topology is finer than the product topology: every open set in the product topology is open in the box topology, but not vice versa. Hence, for any collection of topological spaces $(X_i,\tau_i)_{i\in I}$, the map given by the identity map on sets will be a continuous map $id: (\prod X_i,\tau_{\textrm{box}})\to(\prod X_i,\tau_{\textrm{prod}})$, but it will fail to be continuous in general in the other direction - and this agrees with what we'd expect from the universal property above.)
Quotients also have a universal property: given a topological space $(X,\tau)$ and a subspace $S\subseteq X$, the quotient space $X/S$ is defined to be the set $X/\sim$, where $x\sim y$ if $x,y\in S$ or $x = y$. Intuitively, you smash all of $S$ together into a point. There's a natural map of sets $\pi_S : X\to X/S$ given by mapping $x$ to the equivalence class $[x]$ of $x$ in $X/\sim$ (of course, you can similarly define a quotient space given any equivalence relation on a topological space). We still have to give $X/S$ a topology. You can describe it explicitly by looking at inverses of $U\subseteq X/S$ under $\pi_S$ and demanding that $\pi_S$ be continuous in the "most natural way," but you could also define the whole beast $X/S$ by the following universal property (which describes in what way the choice of topology making $\pi_S$ continuous is the most natural way):

Given $(X,\tau)$ and $S\subseteq X$ as above, the quotient space $X/S$ (if it exists) is a topological space $(X/S,\tau_{quot})$ along with a continuous surjective map $\pi : X\to X/S$ such that for any continuous map $f : X\to Y$ such that $S$ is mapped to a single point under $S$, $f$ factors uniquely as $f = \tilde{f}\circ\pi$, with $\tilde{f} : X/S\to Y$ continuous.

The best way to understand these beasts is to look at many examples. You might start by trying to verify that the explicit descriptions given to you actually satisfy the universal properties I've stated, or by trying to generalize the universal property of the quotient space I've given (it's not the most general definition you can make). If you are ever asked to show something is a quotient or product, a way to do it is to show that the object satisfies the correct universal property: essentially by definition, these properties classify objects uniquely up to unique isomorphism (which is stronger than just up to isomorphism!).
