[Math] the definition of local and global properties and how are they different

general-topology

In brief, I am asking for two things:

  1. A rigorous definition of what "global property" means.

  2. More information on how that differs from (is a larger category than) a local property that is true everywhere within a space. As part of this question: is a local property that holds everywhere a global property of the space? If so, are there local properties that are not also global?

Here is my thinking/ motivation: definitions of local properties abound online, such as in the answer to this question asking that very thing, which says

Not only should the property be true for a neighborhood of each point.
It must also be the case that having a neighborhood with the given property around each point implies that the entire space satisfies the property (satisfying a property locally is not the same as the property being local).

That definition makes me think a local property must hold true everywhere within the space, and therefore also be a global property. This thought process is the origin of my second question, in which I'm confused about the difference in practice between local and global properties.

With respect to my first question, I've spent about two hours searching online and in textbooks (Kasriel's Undergraduate Topology and Kreyszig's Differential Geometry) without finding any definition of "global property". Although it's intuitively pretty clear what the term means, my confusion in my second question suggests my intuition is not complete and I suspect having a rigorous definition of "global property" would help.

Best Answer

A global property is a property that is true for the whole space, when viewed as a single neighborhood. For the second part of your question, a local property that holds everywhere is not necessarily a global property. The easiest example of this is the two-sphere $S^2$. It has the local property of being isomorphic to $R^2$, in that any proper open neighborhood of $S^2$ is in bijection with some open neighborhood in $R^2$, but not the entire space. The best way to view this is the stereographic projection of $S^2$ onto $R^2$. Imagine putting the two sphere at the origin of the plane, so the south pole is on the surface at (0,0) and the north pole is directly above that. To find where a point on $S^2$ maps into the plane, create a line in three-dimensional space through the north pole of the sphere and the given point. Wherever it intersects the plane is the given point in $R^2$. Mapping bottom half of the sphere maps to the unit disk in the plane, and the top half maps to the rest of the plane. But where does the north pole map? It can't map to anywhere on the plane since every point in $R^2$ has an associated point on the sphere. But with any proper neighborhood in $S^2$, there is always at least a point not included in it, take this to be your north pole and you see you map directly into $R^2$. So the two-sphere has the local property of being isomorphic to the plane(in a topological sense) while the entire space as a whole does not, so it is a local property and not a global property.

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