The definition of locality seems to be arbitrary, am I correct?

I keep encountering this term locality. In the context of manifolds it's actually integral to the definition since the manifold can be globally non-Euclidean but must be locally Euclidean. So at what point does a neighborhood cease to be local? Is there a hard cut off or could you define locality in a way that is convenient for you?

• When we say something is true locally, it means that there exists a neighborhood in which it is true. That neighborhood can be different for each point. Commented Dec 20, 2021 at 18:31
• Ok and this is where my confusion comes in because to me it feels circular. So lets say we are using an open ball as a neighborhood. At what point does the radius of our neighbourhood become non-local? Is that a definition we can tweak to a particular application? Commented Dec 20, 2021 at 19:29
• I think you misunderstand. The point is, if there is ANY radius for which it is true, we say that it is locally true. Commented Dec 20, 2021 at 19:30
• To phrase Don’s point another way: locality is a property of points not a property of neighborhoods. Commented Dec 20, 2021 at 19:36
• A couple examples may also be useful: (1) On the sphere, the only open set that is not Euclidean is the entire sphere. If even a single point is not included in an open set, stereographic projection from that point provides a homeomorphism between it and an open set in the plane. (2) On the torus, an open set is Euclidean if it doesn't contain any closed curves that are not contractible. That is, if it doesn't extend completely around the torus in either direction. Commented Dec 22, 2021 at 1:11

There is at least one different point of view on the question. More frequently, you say that a property $$\mathcal P$$ holds locally at a point $$x$$ of a topological space $$X$$ if for every neighbourhood $$U$$ of $$x$$ there is a neighbourhood $$V$$ of $$x$$ such that $$V\subset U$$ and $$V$$ enjoys $$\mathcal P$$. In other words, if there is a neighbourood basis $$\mathcal U$$ at $$x$$ such that every element of $$\mathcal U$$ enjoys $$\mathcal P$$. This is the case for local connectivity, local path-connectivity and many others.
In some, less frequent, cases (for instance, local compactness) this in fact has the other meaning, that is, there is a neighbourhood of $$x$$ enjoying $$\mathcal P$$. But that's definitely rarer.