1. **Problem statement:** We are given the set $\mathbb{Z}$ of integers with the finite closed topology and the set $E$ consisting of all even integers. We need to find the limit points of $E$ in this topology. 2. **Recall definitions:** - A topology is called finite closed if every finite set is closed. - A limit point $x$ of a set $A$ is a point such that every open neighborhood of $x$ contains at least one point of $A$ different from $x$ itself. 3. **Properties of finite closed topology on $\mathbb{Z}$:** - Since finite sets are closed, their complements are open. - Singletons $\{x\}$ are finite and thus closed. - Therefore, every point is isolated because $\{x\}$ is closed, so $\mathbb{Z}$ is a discrete space. 4. **Implication for limit points:** - In a discrete space, no point is a limit point of any set because we can take the singleton $\{x\}$ as an open neighborhood containing only $x$ and no other points. 5. **Conclusion:** - The set $E$ of even integers has **no limit points** in the finite closed topology on $\mathbb{Z}$. **Final answer:** The set of limit points of $E$ is the empty set $\varnothing$.