If [itex]X[/itex] is a topological vector space and [itex]Y[/itex] is a subspace, we can define the quotient space [itex]X/Y[/itex] as the set of all cosets [itex]x + Y[/itex] of elements of [itex]X[/itex]. There is an associated mapping [itex]\pi[/itex], called the quotient map, defined by [itex]\pi(x) = x + Y[/itex]. If I'm not mistaken, there is an equivalence relation lurking here, too: [itex]x \sim y[/itex] iff [itex]\pi (x) = \pi(y)[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Here's my question: We know that if [itex]f[/itex] is some function, then [itex]x\in f^{-1}(A)[/itex] if and only if [itex]f(x) \in A[/itex]. This is fine - the object on the left of the [itex]\in[/itex] is a point, and the object on the right is a set. But if one tries to apply this to the quotient map and a subset [itex]V\subset X[/itex], we have [itex]x \in \pi^{-1}(\pi(V)) [/itex] iff [itex]\pi(x) \in \pi(V)[/itex]. The object on the left of the [itex]\in[/itex] here is a set; the object on the right is a set. So what the heck is this supposed to mean? Did the [itex]\in[/itex] turn into a [itex]\subset[/itex] somehow?

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# Confusion about quotient spaces

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