# Curved spaces that locally aren't flat

martinbn
I understand that as referring to a single ##y##. That is clearly what is described in the SE link.

Note that the Wikipedia definition of Surjective says “if for every ...” instead of “if for any...”

https://en.m.wikipedia.org/wiki/Surjective_function
Well, the same wiki article has this
Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.
Does it refer to a single function or all functions?

lavinia
martinbn
I understand that as referring to a single ##y##. That is clearly what is described in the SE link.
The SE link says that it depends on the context and gives an example. It doesn't say that for "if.., then.." statements it always means one single and not every.

Dale
Mentor
Well, the same wiki article has this

Does it refer to a single function or all functions?
That refers to all functions.

martinbn
That refers to all functions.
Why do you have a problem with the other cases?

weirdoguy
lavinia
Gold Member
It seems that a foundational idea that leads to General Relativity is that in a small enough free falling coordinate system the deviation from flat i.e. the Minkowski metric is too small to be measured. And if one tries more sensitive measuring instruments, one can take a smaller coordinate frame where the instruments can no longer detect the deviations. This would seem to be what is meant by locally flat. Mathematically, this can be interpreted as trajectories of free fall are geodesics with respect to a Levi-Civita connection on a Lorentz metric. Then the deviation from Minkowski in normal coordinates will be of second order. But this is no way means that the curvature tensor is locally zero.

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Dale
Dale
Mentor
Why do you have a problem with the other cases?
I don’t have a problem with it. English has lots of weird rules. I just accept them.

If you want to understand “if for any” as referring to the entire set then you can think of it as saying test the entire set and if at least one element meets the condition then ... It does not mean to test the entire set and if all elements meet the condition then ... That is why the claim fails.

This is the difference between testing the entire set and combining the results with “or” (if for any) vs with “and” (if for every).

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