Let $ (X, \tau) $ be a topological space. We will say that $ X $ has the property $ \mathcal{M} $ if every open coverage of $ X $ admits a finite refinement consisting of connected sets.
Show that if $ X $ has the property $ \mathcal{M} $, then $ X $ is locally connected.
Proof: Let $ U \subset X $ open with $ x \in U $ arbitrary, let us take $ \{U \} $ the open coverage, that by hypothesis there is a finite refinement of connected sets, that is: $x \in \bigcup_{i = 1}^{n} C_i \subset U$ with $ C_i $ connected for each $ i = \{1, \cdots, n \} $, since the finite union of connected is connected, then $ X $ is locally connected.
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