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Remark 20.23.5. This means that if we have two total orderings $<_1$ and $<_2$ on the index set $I$, then we get an isomorphism of complexes $\tau = \pi _2 \circ c_1 : \check{\mathcal{C}}_{ord\text{-}1}(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}_{ord\text{-}2}(\mathcal{U}, \mathcal{F})$. It is clear that

\[ \tau (s)_{i_0 \ldots i_ p} = \text{sign}(\sigma ) s_{i_{\sigma (0)} \ldots i_{\sigma (p)}} \]

where $i_0 <_1 i_1 <_1 \ldots <_1 i_ p$ and $i_{\sigma (0)} <_2 i_{\sigma (1)} <_2 \ldots <_2 i_{\sigma (p)}$. This is the sense in which the ordered Čech complex is independent of the chosen total ordering.

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