In brief, I am asking for two things:
A rigorous definition of what "global property" means.
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.