Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?

A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a continuous map $f:C\to X$ such that $f(c_0) = x$ and $f(c_1) = y$.

Is there also a Hausdorff space satisfying the above?