Is there a topological space $(C,\tau\_C)$ and two points $c\_0\neq c\_1\in C$ such that the following holds?



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Is there also a Hausdorff space satisfying the above?



Connectedness in the language of path-connectedness
No such space $C$ can exist. 


We will derive a contradiction from the assumption that $C,c\_0,c\_1$ as desired exists. 


Let κ be any cardinal greater than $|C|$. View $\kappa$ as an ordinal. For each β in κ add a copy of the unit interval between β and β+1, and add a point ∞ at the end. The resulting "Very Long Line" $L\_\kappa$ is dense and complete, hence connected as a topological space. 


As $L\_\kappa$ is connected, there must be a continuous map $f$ from $C$ into $L\_\kappa$ whose image $L'$ contains $0$ and $\infty$. Let $b\notin L'$. Then the map $h$ that sends everything below $b$ to $0$ and everything above $b$ to $1$ is continuous from $L'$ to the discrete space $\{0,1\}$. 


So the map $h\circ f:C\to \{0,1\}$ witnesses that $\{0,1\}$ is connected, a contradiction. 


(This is just a variant of Helene Sigloch's earlier argument.)


