For which uncountable sets $X$ is it true that there exist a metric $d$ on $X$ such that $(X,d)$ is connected ?

[ The motivation for this question is : I wanted to characterize function $f : X \to X$ such that for any topology $\tau_1,\tau_2$ on $X$ , $f:(X,\tau_1)\to (X,\tau_2)$ is continuous ; I noticed that taking discrete topology in co-domain and the indiscrete topology in domain , since every singleton is open in discrete topology and w.r.t. indiscrete topology $X$ is connected , so such an $f$ must be constant ; then I noticed that if I only required $f$ to be continuous w.r.t. any two metric topologies , then I would get the same conclusion that $f$ is constant , given there exist a metric which makes $X$ connected ]

  • $\begingroup$ very interesting question. Is there a cardinal $\kappa$ such that no connected space of size $\kappa$ is metric? $\endgroup$ – Forever Mozart Jun 16 '16 at 22:21

It’s true precisely for sets $X$ such that $|X|\ge 2^\omega=\mathfrak{c}$. This answer shows that the condition is necessary. To see that it’s sufficient, let $\kappa=|X|\ge 2^\omega$, and fix $p\in X$. Since $\kappa=2^\omega\cdot\kappa$, there is a partition $\{Y_\xi:\xi<\kappa\}$ of $X\setminus\{p\}$ into $\kappa$ sets of cardinality $2^\omega$. For $\xi<\kappa$ let $X_\xi=\{p\}\cup Y_\xi$, and let $f_\xi:X_\xi\to[0,1]$ be any bijection such that $f_\xi(p)=0$. Now define a metric $d$ on $X$ as follows: for any $x,y\in X$,

$$d(x,y)=\begin{cases} |f_\xi(x)-f_\xi(y)|,&\text{if }x,y\in X_\xi\\ f_\xi(x)+f_\eta(y),&\text{if }x\in X_\xi,\,y\in X_\eta,\text{ and }\xi\ne\eta\;. \end{cases}$$

It’s a straightforward exercise to verify that $d$ is a metric. In fact $\langle X,d\rangle$ is the hedgehog of spininess $\kappa$. Since each spine $X_\xi$ is connected, being a copy of $[0,1]$, and the spines all have the point $p$ in common, $X$ is clearly connected.

  • $\begingroup$ oh, yes, of course! $\endgroup$ – Forever Mozart Jun 16 '16 at 22:27

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