The says that the inclusion of the subcomplex generated by singular simplices whose images lie wholly in $U$ or wholly in $V$,
$$
C_*^{\{U,V\}}(M)\hookrightarrow C_*(M),
$$
is a chain-homotopy equivalence. Repeated <barycentric subdivision> supplies the inverse up to chain homotopy.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 4 / b / Solution

For an open cover $\mathcal U$ of $X$, singular chains generated by simplices lying inside one member of $\mathcal U$ form a subcomplex chain-homotopy equivalent to the full singular chain complex. Repeated barycentric subdivision makes chains small.


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 Incoming links: Small simplex theorem