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I'm looking for two references:

  1. A metrizable topology is entirely determined by convergence of sequences
  2. The product topology, if metrizable, is the unique topology on the product space for which we have coordinatewise convergence $x_n=(x_{1,n},x_{2,n},...) \to_n x=(x_1,x_2,...) \iff x_{i,n} \to_n x_n \, \, \forall i\in\mathbb{N}$
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    1. is clear since $A$ is closed iff a convergent sequence $(x_n) \subset A$ has its limit in $A$. 2. Arbitrary product are not metrizable usually. For reference you can look in Munkres, Topology. – 2017-01-02
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    In regards to 2, I meant in the case of a countable product of metrizable topological spaces such that the product space is metrizable. – 2017-01-02
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    I need 2 in the case of $\mathbb{R}^\infty$ such that if I construct a metric of coordinatewise convergence, then the induced topology is the product topology. – 2017-01-02
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    Ok. Well uniqueness is clear by point $1$. So you just need to show that if a product is metrizable then your property is true. And this property is implied by definition of product space. – 2017-01-02
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    @JohnSnow: The product of countably many metrizable spaces is always metrizable. – 2017-01-02
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    Okay thanks that makes sense - If you know a book where these considerations are proven I would like to know it. – 2017-01-02
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    @BrianM.Scott Yes, I found that result in Topology by Dugundji. I was hoping for a reference of 1 and 2, but as N.H. pointed out the proofs are rather trivial, hence they are probably not mentioned verbatim in any book. – 2017-01-02
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    @JohnSnow: The spaces whose topologies are completely determined by their convergent sequences are the [*sequential spaces*](https://en.wikipedia.org/wiki/Sequential_space). They were first systematically studied by Stan Franklin in a pair of papers from 1965 and 1967; there are links to PDFs of both at the foot of the Wikipedia article to which I linked. [Here](http://matwbn.icm.edu.pl/ksiazki/fm/fm57/fm5717.pdf) is a direct link to a PDF of the first paper. – 2017-01-02
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    @JohnSnow : I hope I did not sound rude : this was not my purpose. And I think everything you asked is probably discussed in details in Munkres, Topology, which is a complete and detailled reference for point-set topology. – 2017-01-02
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    @N.H.: Munkres is a good general text, but it is certainly not complete: there’s a great deal of point-set topology that Munkres does not cover. Some of it can be found in Ryszard Engelking’s *General Topology*. Kuratowski’s two-volume *Topology* probably has even more. – 2017-01-02

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  1. is a basic result. A topology on $X$ is uniquely determined by it closure operator, that maps $A\subset X$ to $\bar A =Cl(A).$ If $X$ is a metric space then $p\in \bar A$ iff there is a sequence $(a_n)_{n \in \mathbb N}$ in $A$ such that $$\{p\}=\cap_{n\in \mathbb N}Cl(\{a_m:m\geq n\}).$$ So if two metric topologies on $X$ have the same set of convergent sequences then they have the same closure operator, so they are equal.

Sorry I cannot give you a reference.