Let $\kappa$ be an infinite cardinal. Does there exist a topology $\tau_{\kappa+1}$ on $\kappa+1$ such that for any topological space $(X,\tau)$ with $|X|=\kappa$ the following statement is true?

$(X,\tau)$ is connected if and only if for all $x_0, x_1\in X$ there is a continous, injective map $f:\kappa+1 \to X$ such that $f(0) = x_0$ and $f(\kappa) = x_1$.

(This is a follow-up to Connectedness in the language of path-connectedness)

Let $\kappa$ be an infinite cardinal. Does there exist a topology $\tau_{\kappa+1}$ on $\kappa+1$ such that for any topological space $(X,\tau)$ with $|X|=\kappa$ the following statement is true?

$(X,\tau)$ is connected if and only if for all $x_0, x_1\in X$ there is a continous, injective map $f:\kappa+1 \to X$ such that $f(0) = x_0$ and $f(\kappa) = x_1$.

(This is a follow-up to Connectedness in the language of path-connectedness)