Let \(X\) be a topological space and \(\mathcal{U}=(U_\alpha)_{\alpha\in A}\) be an arbitrary open cover of \(M\). Then a partition of unity subordinate to \(\mathcal{U}\) is an indexed family \((\psi_\alpha)_{\alpha\in A}\) of continuous functions \(\psi_\alpha\colon X\to \mathbb{R}\) with the following properties:
- \(\psi_\alpha(x)\in [0,1]\) for every \(x\in X\).
- \(\supp \psi_\alpha\subset U_\alpha\) for each \(\alpha\in A\).
- The family of supports \((\supp \psi_\alpha)_{\alpha\in A}\) is locally finite, i.e. for every point exists a neighbourhood which intersects with this family only with finitely many \(\alpha\).
- \(\sum_{\alpha\in A} \psi_\alpha(x) = 1\) for all \(x\in X\).
Remarks
- Such a partition of unity exists for all smooth manifolds with or without boundary [1, Theorem 2.25]
- Note, that \(\psi_{\alpha}\) may also be zero, which can be the case if the covering multiplicity is not finite.
References Link to heading
- J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.