This chapter explores the concept of "Countable Topological Spaces" in detail, focusing on how axioms are affected by open and closed sets. It emphasizes the importance of understanding separation properties to determine appropriate proof techniques. The chapter discusses T1-spaces, where points are closed, and the inverse image of points via a continuous function is a closed set. It concludes that countability axioms are properties requiring the existence of a countable set with specific characteristics. - **Countable Spaces**: Spaces with a countable basis or countable local base. - **First Countable Spaces**: Spaces where each point has a countable local base. - **Second Countable Spaces**: Spaces with a countable basis. - **Separable Spaces**: Spaces with a countable dense subset. - **Lindelof Spaces**: Spaces where every open cover has a countable subcover. - **Definition**: A topological space is first countable if each point has a countable local base. - **Theorem**: Every second countable space is a Lindelof space. - **Theorem**: Every second countable space is separable. - **Definition**: A T1 space is one where singleton sets are closed. - **Definition**: A regular space is a T1 space where for each point and closed subset, there exist disjoint open sets containing the point and subset. - **Definition**: A normal space is a T1 space where disjoint closed subsets can be separated by disjoint open sets. - **Theorems**: - Metric spaces are normal. - T1 spaces are regular if and only if every open set containing a point has an open neighborhood disjoint from the point. - T1 spaces are normal if and only if disjoint closed subsets can be separated by disjoint open sets. - Compact Hausdorff spaces are regular and normal. - Continuous functions in normal spaces can map disjoint closed subsets to distinct points. The chapter provides a comprehensive overview of countable and first-countable topological spaces, their properties, and the distinction between regular and normal spaces. It includes detailed proofs and examples to illustrate the concepts.This chapter explores the concept of "Countable Topological Spaces" in detail, focusing on how axioms are affected by open and closed sets. It emphasizes the importance of understanding separation properties to determine appropriate proof techniques. The chapter discusses T1-spaces, where points are closed, and the inverse image of points via a continuous function is a closed set. It concludes that countability axioms are properties requiring the existence of a countable set with specific characteristics. - **Countable Spaces**: Spaces with a countable basis or countable local base. - **First Countable Spaces**: Spaces where each point has a countable local base. - **Second Countable Spaces**: Spaces with a countable basis. - **Separable Spaces**: Spaces with a countable dense subset. - **Lindelof Spaces**: Spaces where every open cover has a countable subcover. - **Definition**: A topological space is first countable if each point has a countable local base. - **Theorem**: Every second countable space is a Lindelof space. - **Theorem**: Every second countable space is separable. - **Definition**: A T1 space is one where singleton sets are closed. - **Definition**: A regular space is a T1 space where for each point and closed subset, there exist disjoint open sets containing the point and subset. - **Definition**: A normal space is a T1 space where disjoint closed subsets can be separated by disjoint open sets. - **Theorems**: - Metric spaces are normal. - T1 spaces are regular if and only if every open set containing a point has an open neighborhood disjoint from the point. - T1 spaces are normal if and only if disjoint closed subsets can be separated by disjoint open sets. - Compact Hausdorff spaces are regular and normal. - Continuous functions in normal spaces can map disjoint closed subsets to distinct points. The chapter provides a comprehensive overview of countable and first-countable topological spaces, their properties, and the distinction between regular and normal spaces. It includes detailed proofs and examples to illustrate the concepts.