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# presheaf of a topological basis

Let $X$ be a topological space and let $\mathcal{B}$ be a basis of its topology. We can regard $\mathcal{B}$ as a category
with objects being the open sets in $\mathcal{B}$ and arrows/morphisms between $U,V\in\mathcal{B}$ to exists only if $U\subset V$, and
where the *only* element of $\mathcal{B}(U,V)$ is the injection map $U\hookrightarrow V$. Let now $\mathcal{C}$ be a complete
category, we now define the
*presheaf of $\mathcal{C}$-objects over the basis $\mathcal{B}$ of the topology of $X$* to be a contravariant functor

$\mathcal{P}:\mathcal{B}\rightarrow\mathcal{C}$ |

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## Mathematics Subject Classification

14F05*no label found*54B40

*no label found*18F20

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