Meaning of the phrase: admits a distance?

Question

I was trying to understand what is a metric space using the well-written wikipedia page. However, in one of the examples, it says:

A geodesic is a shortest possible path between any two of its points. A geodesic metric space is a metric space which admits a geodesic between any two of its points.

What is the meaning of the phrase: "admits a geodesic"? More specifically, what is the meaning of admits here?

Answer 1

"Which admits" is more or less synonymous with "where there exists". In this case, for instance, a geodesic metric space is a metric space where there exists a geodesic between any two points.

The way I think about the language in the "which admits" phrasing (which might be different from how others think about it) is that we imagine the space as a conscious thing, and we ask it "Do you admit geodesics?" and then the metric space answers "Yes, I do admit geodesics." See, for instance, the first Merriam-Webster entry for "admit":

to allow scope for : PERMIT

• admits no possibility of misunderstanding

As opposed to the "where there exists" language, where the metric space is a more passive entity that just lies there to be examined.