Understanding intuitively the interplay between quotient spaces and product spaces.

Question

Hatcher said that the interplay between quotient spaces and product spaces is given in the following proposition:

My questions are:

1- I can not see how this is the interplay between quotient spaces and product spaces, could anyone explain this for me, please?

2-what is the importance of introducing the third space $Z$ and what is the importance of multiplying $f$ by $\mathbb{1}$?

3-This is a statement added after this proposition by Hatcher which is "this can be applied when $Z = I$ to show that a homotopy defined on a quotient space is continuous " but I do not understand the details of this statement, could anyone explain this for me, please?

Answer 1

1. $f:X \to Y$ is a quotient map, or otherwise put, $Y$ is a quotient space of $X$. The statement implies that $Y \times Z$ is also a quotient space of $X \times Z$. So when $Z$ is "nice", a quotient space relationship is preserved by a product operation of multiplying by $Z$. This is unusual, as the quotient topology is defined as a finest topology obeying certain properties (a so-called final topology) while the product topology is defined as the coarsest (smallest) topology beying certain properties (ab initial topology). And examples exist where such operations do not interact well at all (quotient and subspaces etc.)

2. It's technically handy to have the identity as the second map (for the proof). It also combines easily:, e.g. if we have two quotient maps $f: X \to Y$ and $f': X' \to Y'$ then we can write $f \times f': X \times Y \to X' \times Y'$ as $(\mathbb{1}_{X'} \times f') \circ (f \times \mathbb{1}_{Y})$ which is then a quotient map as the composition of two quotient maps, provided that $X'$ and $Y$ are "nice", so that the products with $\mathbb{1}$ are quotient.

3. And if we have a homotopy $H: X \times I \to X$ and $q: X \to X/{\sim}$ is quotient (say for some equivalence relation $\sim$ on $X$, and denoting the class of $x$ by $[x]$ etc.), we know that the homotopy on the quotient defined by $H'([x], t) = [H(x,t)]$ (if this is well-defined, so that $H$ plays well with the relation $\sim$ !) is continuous, as $$H' \circ (q \times \mathbb{1}) = q \circ H$$ and $q \times \mathbb{1}$ is quotient ($I$ is locally compact Hausdorff so "nice"), using the universal property of quotient maps.