Example 3

Let B be a topological space, $\mathscr{B}$ a base of the topology of B$,\mathscr{F}$ a presheaf on B relative to $\mathscr{B}$ and $\mathscr{L}$ a sub-presheaf of $\mathscr{F}$. Then the set $X_{\mathscr{L}}$ is contained in the set $X_{\mathscr{F}}$ and the equivalence relation $R_{\mathscr{F}}$ induces in $X_{\mathscr{L}}$ the equivalence relation $R_{\mathscr{L}}$. The B-morphism $E(i): E_{\mathscr{L}}\rightarrow E_{\mathscr{F}}$ associated with the canonical morphism $i:\mathscr{L}\rightarrow \mathscr{F}($I, p. 48, example 2) is therefore injective. Since $E_{\mathscr{L}}$ and $E_{\mathscr{F}}$ are etale B-spaces, the mapping $E(i)$ is open and even etale (I, p. 30, cor. 1), and therefore induces a homeomorphism of $E_{\mathscr{L}}$ onto an open subset of $E_{\mathscr{F}}$.