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Suppose we are asked to prove that the quotient space $\mathbb R/\mathbb Z$ of $\mathbb R$ equipped with the quotient topology is compact. Has this question provided enough information for us to answer it? Do we not need to know the topology given to $\mathbb R$? I ask this because we can attach a wide variety of topologies to $\mathbb R$.

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    I'd like a little clarification, as there are two possible meanings of $\mathbb R / \mathbb Z$. One is that this term means the quotient group of the additive group $\mathbb R$ by the additive subgroup $\mathbb Z$; this group is often called the circle group, and is isomorphic to the multiplicative group of complex numbers of modules 1, and is compact. The other is that the symbol $X/A$ means $X$ with $A$ shrunk to a point; in this case the result looks non compact!2012-05-13

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You may generally assume that any $\Bbb R^n$ has the usual (Euclidean) topology unless some other topology is explicitly specified or made very clear by the context.