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Prove that a topological product of uncountably many metric spaces, each having more than one element, is never metrizable.

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    Why downvote? Is statement incorrect? – 2017-02-22
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    no it is correct, I need it to be proven – 2017-02-22
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    What have you tried? Where did you get stuck? Please don't just ask us to do your homework for you. – 2017-02-22
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    I am sorry about not telling you where I got stuck. I said since I is not countable there will be infinietly many elements j which U_j not equal to X_j – 2017-02-22
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    I am sorry about not telling you where I got stuck. I said since I is not countable there will be infinietly many NBHD basis . And since the product of infinitely metric space has to have a countable NBRH basis that will contradict I so that will make the product of infinitely metric space not metrizable but I am not convinced – 2017-02-22

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A basic open set is $$\prod U_i$$ where $U_i\subseteq X_i$ is open and for all but finitely many $i$, $U_i=X_i$.

Now let $O_n$ be a countable number of basic open sets and assume that $$\bigcap O_n\neq \emptyset$$

Then there is an index $i$ such that for all $n$ the $i$th coordinate of $O_n$ is $X_i$. Since $X_i$ has more than one element we see that $\bigcap O_n$ also must have more than one element. This implies that the space is non metrizable for in a metric space $$\{x\}=\bigcap B_x(\frac{1}{n})$$

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    Thank you but I still have a question .Does the NBHD here countable or not? if it is not how to prove it – 2017-02-22