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$\mathbf {The \ Problem \ is}:$

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$\mathbf {My \ approach}:$ Actually, I tried to show $T = \frac{F=(X×W)\sqcup (Y×W)}{(f(a),w)\sim (g(a),w)}$ for all $a\in A$ and $w\in W$ is homeomorphic to $Z×W$ using the fact that $Z= \frac{D=X\sqcup Y}{f(a)\sim g(a)}$ for all $a\in A.$

Firstly, I defined $q : F \to Z×W $ by $(r,w) \mapsto ([r],w)$ ,then q is well-defined and continuous(as component wise continuous) .

Then, we can apply universal property to extend $q$ to $Q$ from $T$ to $Z×W$ as $q(f(a),w)=q(g(a),w)$ for all $a$ , all $w.$

Then $Q$ is one-one,onto, continuous but I am getting confused in showing $Q$ is open map .

Taking an open set $P$ in $T$ , then we can write $P$ as an union of all those $[(x,w)]$ such that $x \notin f(A)$, and those $[(y,w)]$ such that $y \notin g(A)$ and those $[(f(a),w)]$ which are inside $P.$

Here all the spaces are topological spaces and all maps are continuous .

After that, I am getting confused . A small hint is warmly appreciated.

Thanks in advance.

Rabi Kumar Chakraborty
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  • A particular case is already answered here try to mimic the idea. – Sumanta May 25 '21 at 09:51
  • dear Sumanta, I know the case when $W$ is a locally compact, hausdorff space ....but when it is not, how can we say that $q×id$ is also a quotient map when $q$ is ? – Rabi Kumar Chakraborty May 25 '21 at 11:11
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    You need to specify which category you're working in. From the context, I guess it's Top, the category of topological spaces and continuous maps. Is that right? Please edit your question to include this information. – Alex Kruckman May 25 '21 at 13:42
  • dear Alex Kruckman, thanks for rectifying the mistake . – Rabi Kumar Chakraborty May 25 '21 at 14:01
  • If $q : A \to B$ is a quotient map, then in general there exist spaces $W$ such that $q \times id_W$ is not a quotient map. See for example https://math.stackexchange.com/q/31697, especially Ronnie Brown's comment. – Paul Frost May 25 '21 at 23:12
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    The claim is false (for the reasons already mentioned). Pushouts are not stable under products in $\mathbf{Top}$. https://ncatlab.org/nlab/show/exponential+law+for+spaces – Martin Brandenburg May 27 '21 at 22:59
  • Respected Martin Brandenburg, can you please give a link to whitehead's lemma related to product of quotient maps (as I searched and found a whitehead's lemma in homotopy theory , not on quotient maps) ? – Rabi Kumar Chakraborty May 29 '21 at 19:19

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