Connected im kleinen

Let $(X,\tau)$ be a topological space and let $x \in X$. We define $X$ to be connected im kleinen at $x$ if for each closed $C \subset X \setminus \{x\}$ there is a continuum $K \subset X$ such that $x \in \mathrm{Int}(K) \subset K \subset X \setminus C$.