"Continuity" in a metric space

Question

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.

Answer 1

I do not agree with Ben's comment that completeness is the right formalization of what you're asking about. For example the Cantor set is also complete but I think it's hard to argue that it has "no gaps"!

I think a better way to formalize "no gaps" is that $\mathbb{R}$ is connected, which informally means that it can't be separated into two pieces. This neatly distinguishes $\mathbb{R}$ from the Cantor set, which is totally disconnected. The connectedness of $\mathbb{R}$ is related to but not the same thing as its completeness and is one of its most important basic properties; for example it immediately implies the intermediate value theorem.

In mathematics we only use the term "continuous" to refer to functions, not to spaces; for spaces we have more precise terms like connectedness and we use those terms instead. The naive distinction between "continuous" and "discrete" breaks down badly; for example already $\mathbb{Q}$ is neither (it is also totally disconnected). So we need more precise terminology.

Even connectedness may not be quite right, since for example $\mathbb{R}^2 \setminus \{ 0 \}$ is also connected and one could argue it has a "gap," namely the hole in the middle. This example is not complete wrt the induced metric, so one could ask for both complete + connected. Unfortunately $\mathbb{R}^2 \setminus \{ 0 \}$ is homeomorphic to the cylinder $S^1 \times \mathbb{R}$, which has a metric making it complete! This metric makes the hole "infinitely far away." One wonders whether the cylinder qualifies as having a "gap" or not; at this point we really need more precise terminology to say what exactly we mean. Spaces can vary in many different ways!