This paper explores countable topological spaces, focusing on first and second countable spaces, separable spaces, and Lindelöf spaces. It discusses the properties of these spaces and how they are affected by open and closed sets. The paper also examines the separation properties of $ T_1 $-spaces, where points are closed, and the inverse image of a point under a continuous function is a closed set. It concludes that the Axiom of Countability is a property of certain mathematical objects requiring the existence of a countable set with specific properties. A topological space $(X, \tau)$ is first countable if for each $x \in X$, there exists a countable local base at $x$. A space is second countable if it has a countable basis. A separable space has a countable dense subset. A Lindelöf space is one where every open cover has a countable subcover. The paper proves that every second countable space is Lindelöf and separable. It also discusses properties of first countable spaces, such as the existence of a countable local base and the characterization of closure points via sequences. The paper defines regular and normal spaces, and proves that every metric space is normal. It also shows that a $ T_1 $-topological space is regular if and only if for each point $x$ and open set $U$ containing $x$, there exists an open set $V$ containing $x$ such that $\bar{V} \subseteq U$. Furthermore, it proves that every compact Hausdorff space is regular and normal. Finally, it uses Urysohn's Lemma to show that in a normal space, there exists a continuous function separating two disjoint closed sets.This paper explores countable topological spaces, focusing on first and second countable spaces, separable spaces, and Lindelöf spaces. It discusses the properties of these spaces and how they are affected by open and closed sets. The paper also examines the separation properties of $ T_1 $-spaces, where points are closed, and the inverse image of a point under a continuous function is a closed set. It concludes that the Axiom of Countability is a property of certain mathematical objects requiring the existence of a countable set with specific properties. A topological space $(X, \tau)$ is first countable if for each $x \in X$, there exists a countable local base at $x$. A space is second countable if it has a countable basis. A separable space has a countable dense subset. A Lindelöf space is one where every open cover has a countable subcover. The paper proves that every second countable space is Lindelöf and separable. It also discusses properties of first countable spaces, such as the existence of a countable local base and the characterization of closure points via sequences. The paper defines regular and normal spaces, and proves that every metric space is normal. It also shows that a $ T_1 $-topological space is regular if and only if for each point $x$ and open set $U$ containing $x$, there exists an open set $V$ containing $x$ such that $\bar{V} \subseteq U$. Furthermore, it proves that every compact Hausdorff space is regular and normal. Finally, it uses Urysohn's Lemma to show that in a normal space, there exists a continuous function separating two disjoint closed sets.