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Fix $n$. Let $A$ be the space of sequences with the elements in the set $\{0,1,2,...,n-1\}$.

Define $h:A\rightarrow \mathbb{R}/\mathbb{Z}$ by $\displaystyle h(\{x_0,x_1,...\})=\frac{x_0}{n}+\frac{x_1}{n^2}+...$. Prove that $h$ is continuous.

$h$ is clearly surjective. For continuity I was thinking of showing that given a neighbourhood at some point $y$ in $\mathbb{R}/\mathbb{Z}$ the preimage also contains a neighbourhood in $A$ around $h^{-1}(y)$. But how do I define a neighbourhood in $A$?

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    I think you mean $h:A\rightarrow \mathbb R/\mathbb Z$? – 2017-01-18
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    Yep, Sorry about that. – 2017-01-18
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    Very likely you should view $A$ as an infinite product of $n$-element discrete spaces and give it the product topology, But I can't say for sure given the limited amount of context you have provided. – 2017-01-18

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