1. **State the problem:** We have a set $X=\{x,y,z,w,t\}$ with topology $\tau=\{X,\emptyset,\{x\},\{x,y\},\{x,z,w\},\{x,y,z,w\},\{x,y,t\}\}$. We need to find the neighborhood system of the point $t$. 2. **Recall the definition:** A neighborhood of a point $t$ is any set $N$ containing an open set $U$ such that $t \in U \subseteq N$. 3. **Identify open sets containing $t$:** From $\tau$, the open sets containing $t$ are $X=\{x,y,z,w,t\}$ and $\{x,y,t\}$. 4. **Find neighborhoods:** Any neighborhood of $t$ must contain at least one of these open sets. So neighborhoods of $t$ are all sets $N$ such that $\{x,y,t\} \subseteq N$ or $X \subseteq N$. 5. **List neighborhoods:** Since $X$ is the whole set, any neighborhood containing $\{x,y,t\}$ is a neighborhood of $t$. The minimal neighborhood is $\{x,y,t\}$ itself. **Final answer:** The neighborhood system of $t$ is all sets $N$ with $\{x,y,t\} \subseteq N \subseteq X$. In other words, neighborhoods of $t$ are $\{x,y,t\}$ and $X$ itself.