#Math101 Session 32

Time：Dec. 18, 2025, 14:00-16:00

Location：E104 Hai Tian Building, Chang'an Campus

Speaker：Mr. Sun, Wenhao (Class of 2024)

Title：Axioms of countability and separation

Abstract：How to tell the “closeness” of points is a basic question in topology, especially in settings where metric notions are not relevant. In this talk, we introduce two fundamental collections of conditions that address this issue: the axioms of countability and the axioms of separation. We begin with the axioms of first countability and second countability, presenting their definitions along with examples and counterexamples. We then introduce the separation axioms, from the minimal T0 condition through T1, Hausdorff (T2), regular, and normal spaces. We explore how these axioms formalize the ability to control the structure of local neighborhoods and distinguish points and sets using open neighborhoods in a general topological space.

（文：申爽/审核：郭千桥）