1. **Problem Statement:** We are given the set $A = \{1, 2, 3, \ldots, 10\}$ as a subset of the integers $\mathbb{Z}$ equipped with the finite closed topology. We need to find the limit points of $A$ in this topology. 2. **Understanding the finite closed topology:** In the finite closed topology on $\mathbb{Z}$, the closed sets are exactly the finite subsets of $\mathbb{Z}$ and the whole set $\mathbb{Z}$. Consequently, the open sets are complements of finite sets, i.e., cofinite sets. 3. **Definition of limit points:** A point $x$ is a limit point of a set $A$ if every open neighborhood of $x$ contains at least one point of $A$ different from $x$ itself. 4. **Analyzing neighborhoods in finite closed topology:** Since open neighborhoods are cofinite sets, any open neighborhood of $x$ contains all but finitely many integers. 5. **Checking if $x$ is a limit point of $A$:** For $x$ to be a limit point of $A$, every cofinite set containing $x$ must contain some point of $A$ different from $x$. 6. **Since $A$ is finite:** $A$ has only 10 elements. For any $x \in \mathbb{Z}$, consider the open neighborhood $U = \mathbb{Z} \setminus (A \setminus \{x\})$. This is cofinite because $A \setminus \{x\}$ is finite. 7. **If $x \notin A$:** Then $U$ contains $x$ but no points of $A$ because $U$ excludes all points of $A$. So $x$ is not a limit point. 8. **If $x \in A$:** Then $U$ contains $x$ but excludes all other points of $A$. So $U$ contains no points of $A$ different from $x$. Hence, $x$ is not a limit point. 9. **Conclusion:** No point in $\mathbb{Z}$ is a limit point of $A$ under the finite closed topology. **Final answer:** The set of limit points of $A$ is the empty set $\emptyset$.