I just learned about metric spaces and their basic properties and definitions, such as the definition of a continuous function that maps two metric spaces. However, when I say "continuity" on a metric space, I am not talking about continuos functions, but rather the continuity of the space itself. Like, how do you characterize the property of a metric space being "continuous", kind of like the real numbers, that is to say, having no gaps. In the real numbers, we usually characterize this with the completeness axiom, however I don't think that can be directly translated into a metric space since there is no least or bigger than. I remember reading that an equivalent formulation of the completeness axiom is the nested interval theorem, so I figured maybe it is possible to create an analogous formulation with metric spaces, instead of nested intervals, nested closed sets. Would that work? I also read about the property of completeness in a metric space, and that is something about cauchy sequences always converging to a point on the space, but I am not sure if that completeness is the same completeness we talk about in the real numbers aka having no gaps. So in conclusion I want to know, what are some characterizations of the "continuity" of a metric space? and also I would like to know if my idea of generalizing the nested interval theorem would work in the desired way. Thanks.
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