Question

prove that the product of two quotient maps needs not be a quotient map.​

Answer 1

Answer:

Step-by-step explanation:

Your weaker statement is almost true.

If f:X→Y

is a quotient map and Z is locally compact, then f×id

is a quotient map. I believe that this result is due to Whitehead.

More generally, if f:X→Y

and g:Z→W are quotient maps and Y and Z are locally compact, then the product f×g:X×Z→Y×W

is a quotient map.

Why? Use the Whitehead theorem twice, since f×g=(id×g)∘(f×id)

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