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?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Confusion about quotient spaces

Loading...

Similar Threads - Confusion quotient spaces | Date |
---|---|

B Proof of quotient rule using Leibniz differentials | Jun 10, 2017 |

B Confusion between ##\theta## and ##d\theta## | Jan 14, 2017 |

B Confusion regarding area of this figure | Nov 5, 2016 |

I Fourier Series: I don't understand where I am wrong -- please help | Jun 16, 2016 |

I Geometric Series Confusion | Apr 19, 2016 |

**Physics Forums - The Fusion of Science and Community**