Minimum number of elements of a connected subset of a metric space

Question

$(X,d)$ is a metric space and $A\subseteq X$ is connected with at least two distinct points. Then what is the cardinality of $A?$

Now I can see that it cannot be anything finite but what kind of infinity? Countable or Uncountable? Can that be decided? If so how?

Thank you.

Answer 1

Suppose $x$ and $y$ are two distinct points of $A$. For every $r \in (0, d(x,y))$ there must be $z \in A$ with $d(x,z) = r$, otherwise we'd have $A = \{z: d(x,z) < r\} \cup \{z: d(x,z) > r\}$ disconnected. So the cardinality of $A$ must be at least $\frak c$ (the continuum).