• if $\{ U_ i \to U\} _{i \in I}$ is a $\tau$-covering, then there exist

1. a $\tau '$-covering $\{ V_ j \to U\} _{j \in J}$,

2. a $\tau$-covering $\{ f_ j : W_ j \to V_ j\}$ consisting of a single $f_ j \in \mathcal{P}$, and

3. a $\tau '$-covering $\{ W_{jk} \to W_ j\} _{k \in K_ j}$

such that $\{ W_{jk} \to U\} _{j \in J, k \in K_ j}$ is a refinement of $\{ U_ i \to U\} _{i \in I}$.

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