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A topological space \((X,\mathcal{T})\) is paracompact if every open cover of \(X\) has a refinement which is locally finite, i.e. given \(\{U_i|i\in I\}\subset\mathcal{T}\) such that \(\cup_{i\in I}U_i=X\), there exists a collection of open sets \(\{V_j|j\in J\}\subseteq\mathcal{T}\) such that \(\cup_{j\in J}V_j=X\), \(\forall j\in J:\exists i\in I:V_j\subseteq U_i\), and \(\forall x\in X:\exists \mathcal{N}_x\in\mathcal{T}:x\in \mathcal{N}_x\land\#\{j\in J|\mathcal{N}_x\cap V_j\neq\emptyset\}\) (a refinement of an open cover is a second cover such that each open set of the refinement is contained in at least one of the sets of the original cover and is locally finite if, for each point in the space, there is some neighbourhood of the point such that a finite number of sets in the refinement intersect the neighbourhood). Paracompactness is a weaker property than compactness, and is often a required property of manifolds as every open cover of a paracompact space admits a partition of unity.