Let $C$ be the category of topological spaces.
Given a topological space $X$, we say a collection $\{f_i : U_i \rightarrow X\}$ of morphisms in $C$ is a "covering family" of $X$ if each $f_i$ is a continuous open embedding (in particular also a homeomorphism onto its image),
and such that the images of the $f_i$'s cover $X$.
(My apologies: I forgot to make the "covering families" actual coverings; I have now added the above line to include this requirement. Thank you to the comments who pointed this out.)
Let $\tau$ be the function on $\mathrm{Obj}(C)$ which takes each object $X$ of $C$ to the set of $C$-sieves $\xi$ on $X$ such that $\xi$ is generated by some covering family of $X$.
I would like to show $\tau$ satisfies the conditions for a Grothendieck topology (Definition 3.1 at the nLab page).
The maximal sieve on $X$ is always in $\tau(X)$, since e.g. it is generated by the open covering consisting of just $\{X \xrightarrow{ \mathrm{id} } X \}$.
"Pullback stability" condition: let $f : Y \rightarrow X$ be a given continuous map, let $\xi \in \tau(X)$ a given $\tau$-sieve; then find a covering family $\{g_i : U_i \rightarrow X\}_{i\in I}$ which generates $\xi$; then, we can show the pullback sieve $f^* \xi$ on $Y$ is generated by the "pullback" covering family $\{ f^*g_i : f^{-1}U_i \rightarrow Y\}$ on $Y$.
"Local character condition": let $\nu$ be a sieve on $X$, and let $\xi \in \tau(X)$ a $\tau$-sieve, and suppose for every $\left(Y\xrightarrow{g}X \right)\in \xi$ that $g^*\nu \in \tau(Y)$. We want to show $\nu \in \tau(X)$.
Since $\xi$ is a $\tau$-sieve, first find $\{f_i : U_i\rightarrow X\}_{i\in I}$ a covering family which generates $\xi$. The given condition implies that for every $i_0 \in I$, for a space $Y$ and a map $g : Y \rightarrow U_{i_0}$, then $g^*f_{i_0}^*\nu \in \tau(Y)$; i.e., there exists a covering $\{h_j : V_j \rightarrow Y\}_{j \in J}$ such that $g^*f_{i_0}^*\nu$ is generated by $\{h_j\}_{j\in J}$.
In particular, for each $i_0 \in I$, take $Y = U_{i_0}$ and $g = \mathrm{id}_Y$; then the above implies there is a covering $\{h_j : V_j \rightarrow U_{i_0}\}_{j \in J}$ such that $f_{i_0}^*\nu$ is generated by the $h$'s.
However, at this point I don't know how to proceed. My first guess is that the last line is saying that the $U_i$ for $i \neq i_0$ cover $U_{i_0}$, and we can take $J = I\setminus \{i_0\}$ and $V^{i_0}_j = U_j \cap U_{i_0}$; then somehow we can show that the collection $\{ (f_i|_{V^i_j} : V^i_j \rightarrow X) \;|\; i,j \in J \,\&\, i\neq j\}$ gives an $X$-covering family that generates $\nu$, hence $\nu$ is a $\tau$-sieve. But unfortunately I'm not sure how to show these steps.
Would anyone have any suggestions on how to approach proving the "local character" condition? (Or, is my assumption that it should hold here incorrect?)
Edit: I guess the fact I am trying to prove for coverages on a general category $C$ is:
If we take the sieves generated from a coverage as covering sieves, these sieves produce a Grothendieck topology.