Fix a set X with at least 2 elements and $p\in X$. Let $\tau_{-p}=\{U\subseteq X|p\notin U\}\cup X$.
So we check the criterias of the definition. Just wanted to know if I have shown it in a correct way.
1) $X,\emptyset$ are clearly containing in $\tau_{-p}$
2). Let $\{U_{I}\}$ be an arbitary subcollection of $\tau_{-p}$. Since $p\notin U_i$ for all $i\in I$, clearly $p\notin \bigcup_i \,U_i$
3). Let $\{U_{I}\}$ be an arbitary subcollection of $\tau_{-p}$. Since $p\notin U_i$ for all $i\in I$, clearly $p\notin \bigcap_i \,U_i$.
I just thought this was very easy to show and wondered if I did it wrong?