Connected im kleinen

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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 $x \in \mathrm{Int}(K) \subset K \subset X \setminus C$, where $\mathrm{Int}(K)$ denotes the interior of $K$.

Properties

Locally connected if and only if connected im kleinen