I'm given the following definition.
A topological space $X$ has property $S$ if and only if every open cover of $X$ can be refined by a cover of finitely many connected sets.
We also have
A space $X$ is connected im kleinen at a point $x$ if for every open neighborhood $U$ of $x$ there is another neighborhood $V$ of $x$ such that any two points in $V$ lie in some connected subset of $U$.
We need to show that if $X$ has property $S$, then it is connected im kleinen at every point. So we first fix an $x \in X$ and some open neighborhood $U$ of $x$. Then we let $\mathscr{U}$ be an open cover of $X\setminus \{x\}$. Then $\mathscr{U}' = \mathscr{U} \cup \{U\}$ is an open cover of $X$. By property $S$, there is a refinement $\{V_1,..., V_n\}$ of connected sets which also covers $X$. Hence $x \in V_i$ for some $i$, moreoever $V_i \subset U$.
To finish off the proof, I need to show that there is an open nhood $V$ of $x$ with $V \subset V_i$ but I'm not sure how to guarantee its existence. Any hints?
If $X$ is Hausdorff let $\mathscr{U}$ be a cover of $X\setminus U$ by open sets whose closures don’t contain $x$, get a finite refinement $\mathscr{C}$ of $\mathscr{U}\cup\{U\}$ by connected sets, let $\mathscr{C}_0=\{\operatorname{cl}C\in\mathscr{C}:x\in\operatorname{cl}C\}$, let $\mathscr{C}_1=\{\operatorname{cl}C:x\notin\operatorname{cl}C\}$, let $K=\bigcup\mathscr{C}_0$, and let $W=X\setminus\bigcup\mathscr{C}_1$. Then $K$ is connected, $W$ is open, and $x\in W\subseteq K\subseteq U$. I don’t at the moment see how to prove it for arbitrary $X$, however, and the original proof by R.L. Moore in ‘Concerning Connectedness im kleinen and a Related Property,’ Fund. Math. 3, 232-237 (1922), is for metric spaces.