Development of a topological space

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Definition

Let $X$ be a topological space. A development for $X$ is a countable collection $F_1,F_2,\dots$ of open covers of $X$, such that for any closed subset $C\subset X$ and any point $p\notin C$, there exists a $F_j$ such that no member of $F_j$ which contains $p$ intersects $C$.

A topological space which has a development is termed a developable space.