Topology
拓扑学是研究几何图形在连续变形下保持不变性质的数学分支，被誉为"橡皮几何"。课程内容包括拓扑空间与连续映射、度量拓扑、积拓扑与商拓扑、连通性与道路连通性、紧致性、可数性公理与分离公理、度量化定理、基本群与覆盖空间初步。学生将学习拓扑学的基本概念和方法，培养几何直觉和抽象思维能力。本课程是代数拓扑、微分几何、泛函分析等高级课程的基础。
Topology studies properties of geometric spaces preserved under continuous deformations, often called "rubber sheet geometry." Topics include topological spaces and continuous maps, metric topologies, product and quotient topologies, connectedness and path connectedness, compactness, countability and separation axioms, metrization theorems, and introduction to fundamental groups and covering spaces.
研究紧致曲面的拓扑分类定理，证明闭曲面的欧拉示性数与亏格的关系。构造不同类型的曲面（球面、环面、射影平面、克莱因瓶），计算它们的欧拉示性数，探讨连通和运算的性质。通过三角剖分方法验证欧拉公式，分析曲面定向性与欧拉示性数的关系，展示拓扑不变量在几何分类中的核心作用。
Study the topological classification theorem for compact surfaces, prove the relationship between Euler characteristic and genus of closed surfaces. Construct different types of surfaces (sphere, torus, projective plane, Klein bottle), compute their Euler characteristics, and explore properties of connected sum operations.