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Let $X = \mathbb{R}$ and let $(X_i)_{i\in \omega}$ be a partition of $X$ into [Bernstein sets](https://en.wikipedia.org/wiki/Bernstein_set) (for the Euclidean topology).
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Given some $x \in X$, there is a unique $n \in \omega$ such that $x \in X_n$. Then $\{x\} \cup (U \cap \bigcup_{i<n}X_i)$, with $U \subseteq X$ with $U$ open in the usual topology, for a neighbourhood basis for $x$.
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Given some $x \in X$, there is a unique $n \in \omega$ such that $x \in X_n$. Then $\{x\} \cup (U \cap \bigcup_{i<n}X_i)$, with $U \subseteq X$ with $U$ open in the usual topology, form a neighbourhood basis for $x$.
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This topology in finer than {S25}.
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