metric characterization for connectedness

Question

Is there a metric characterization of connectedness? I'm looking for something like the following metric characterization of compactness: A metrizable topological space is compact if, and only if, every metric inducing the topology is complete and totally bounded.

So, is there any property $P$ of a metric such that a metrizable topological space is connected if, and only if, every metric inducing the topology has property $P$.

If no such $P$ exists, is there a property $P$ such that every meteric space satisfying it is connected, and a metrizable topological space is connected provided there is some metric inducing the topology and which satisfies $P$.

Answer 1

There is for compact spaces:

A compact metric space $X$ is connected if and only if for all $a,b\in X$ and all $\varepsilon>0$ there exist points $a=p_0, p_1, \ldots, p_k=b$ such that $d(p_i,p_{i+1})<\varepsilon$.

And it's not hard to prove this. It's difficult for me to imagine there's some general characterization given examples of connected spaces like the Knaster–Kuratowski fan.

Answer 2

Let us first consider a different characterization of compactness: a metrizable space is compact iff the range of every metric is compact. Necessity follows from continuity of the metric, and sufficiency can be obtained from the fact that every non-compact metrizable space admits an unbounded metric.

This suggest the analogous statement: a metrizable space is connected iff the range of every metric is connected. This turns out to be true, and even simpler.

Necessity again follows from continuity of the metric. To show sufficiency, let $(X, \rho)$ be a disconnected metric space and let $A, B$ be a separation of $X$. We can define a topologically equivalent metric by $$ \sigma(x, y) = \cases{ \min\{ 1, \rho(x, y) \} & if $x,y \in A$ or $x, y \in B$,\\ 2 & otherwise. } $$ Clearly the range of $\sigma$ is not convex, since it contains 0 and 2, but not 3/2, therefore it is disconnected.

Corollary: a metrizable space is a continuum iff the range of every metric is a closed bounded interval.

P.S.: There is a question you did not ask, but which is suggested by your example: is there a property $P$ of metrics such that the existence of a metric with property $P$ implies connectedness and connectedness implies that all metrics have property $P$. Equivalently: is there a property $P$ such that a metric space is connected iff its metric has property $P$. I don't have an answer to that.