Franklin's Notes

Spatial topos

Given a topological space $(I,\Theta)$, it turns out that the category of sheaves $\mathsf{Top}(I)$ is always a topos , and is called a spatial topos. We have the following data for the subobject classifier of this topos:

Notice that we can define a partial ordering on the germs $[U]{\beta(x)}$ by letting $[U]{\beta(x)}\sqsubset [V]{\beta(x)}$ if and only if there is a neighborhood $W$ of $q(x)$ in which the restriction of $U$ is contained in the restriction of $V$, or such that $W\cap U\subset W\cap V$. That is, we say that $[U]{\beta(x)}\sqsubset [V]_{\beta(x)}$ if "points of $U$ sufficiently close to $q(x)$ are contained in $V$".




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