求推荐含大量实例的拓扑群及应用入门书籍，咨询核心学科

Hey there! Since you already have a solid foundation in general topology (via Munkres) and group theory, you're perfectly positioned to dive into topological groups. Let's tackle your questions one by one:

Recommended Introductory Books with Plenty of Examples #

These picks are tailored for someone at your level, packed with concrete instances to make abstract concepts tangible:

• 《Topological Groups: A First Course》by Sidney A. Morris
  This is hands-down my top recommendation for beginners. It's explicitly designed for readers who already know general topology and group theory, and every core concept is paired with relatable examples—think the additive group of real numbers, matrix groups like (GL(n,\mathbb{R})), p-adic number groups, and more. It avoids overly dense abstraction and focuses on building intuition through examples.
• 《Topological Groups》by John G. Hocking
  Another great entry point that aligns well with Munkres' topological style. It uses familiar topological spaces (from your Munkres studies) to frame group structures, making the transition to topological groups feel natural. The book is loaded with illustrative examples that connect the two disciplines you already know.

Core Subjects to Support Your Topological Groups Studies #

Beyond your existing background, these areas will be invaluable as you deepen your understanding and explore applications:

• Functional Analysis
  When you move into applications like representation theory of topological groups, you'll need basics of Banach spaces, Hilbert spaces, and linear operators. This is especially critical for topics like unitary representations of locally compact groups.
• Differential Geometry & Lie Groups
  A huge subclass of topological groups are Lie groups (smooth topological groups), which are ubiquitous in physics, algebra, and geometry. Knowing the basics of smooth manifolds, Lie algebras, and their connections to Lie groups will let you explore rich, practical examples and applications.
• Measure Theory & Harmonic Analysis
  Haar measure (a canonical measure on locally compact topological groups) is a foundational tool, and understanding it requires measure theory fundamentals. Harmonic analysis builds on this, with applications ranging from signal processing to number theory—all rooted in topological group structures.
• Algebraic Topology
  While you know general topology, adding basics of fundamental groups, homology, and cohomology will help you analyze the topological properties of groups (like connectedness, compactness) from an algebraic perspective. Many classification results for topological groups rely on these algebraic topology tools.

Hope these recommendations and insights set you up for success with topological groups!

内容的提问来源于stack exchange，提问作者Sumit Mittal