Introduction to Metric and Topological Spaces

Citation

Sutherland, W. A. (2009). Introduction to Metric and Topological Spaces (2nd ed.). Oxford University Press.

Chapter Summary

The book has 17 short chapters in roughly 200 pages (catalogues give 206 or 224). The publisher's description states the plan: move gradually from familiar real analysis to abstract topological spaces, with metric spaces as the bridge and continuity as the motivating concept. Several concepts are introduced first for metric spaces and then repeated for topological spaces, so the second version arrives with familiar examples already in hand. The second edition replaces the 1975 Clarendon first edition and adds a companion website.

Chapters 1–4: Introduction; Notation and terminology; More on sets and functions; Review of some real analysis

Preparatory chapters. They fix the notation and the set and function language used later, and review the real analysis on which the later chapters build.

Chapters 5–6: Metric spaces; More concepts in metric spaces

Introduce metric spaces and the first topological ideas in that setting, extending the analytic notions of the previous review to a general distance function.

Chapters 7–9: Topological spaces; Continuity in topological spaces; Some concepts in topological spaces

The move to topological spaces. Continuity is restated without distances, and notions already seen for metric spaces are repeated in the more general setting.

Chapters 10–11: Subspaces and product spaces; The Hausdorff condition

Ways of building new spaces from old ones, and a separation condition on topological spaces.

Chapters 12–14: Connected spaces; Compact spaces; Sequential compactness

The first two of the trio of "connectedness, compactness and completeness" that the publisher says the discussion develops. Compactness appears in two chapters, one on compact spaces and one on sequential compactness.

Chapter 15: Quotient spaces and surfaces

The geometric side of the subject: the publisher says the book describes some standard surfaces, the Möbius bands, doughnuts and Klein bottles of popular accounts, and shows how they fit into the same story. The chapter title pairs quotient spaces with surfaces.

Chapters 16–17: Uniform convergence; Complete metric spaces

A return to analytic topics: uniform convergence of functions, and completeness of metric spaces, the third member of the trio.

Key Concepts

Chapters 1–6: From Real Analysis to Metric Spaces

• Metric space: a set with a distance function, the setting in which limits and continuity are first generalised.
• Continuity: the guiding notion, first as a statement about distances and later in the language of open sets.

Chapters 7–11: Topological Spaces

• Topological space: a set with a family of open sets, which keeps continuity and convergence without any distance.
• Subspace and product space: new topologies induced on subsets and on products.
• Hausdorff condition: a separation property that rules out points that cannot be told apart by open sets.

Chapters 12–17: Connectedness, Compactness, Completeness

• Connected space: a space that cannot be split into two disjoint open sets.
• Compact and sequentially compact space: the finiteness conditions behind results such as the existence of extrema.
• Quotient space: a space obtained by gluing points together, used to construct surfaces.
• Uniform convergence and complete metric spaces: convergence of functions at a uniform rate, and spaces where every Cauchy sequence converges.

Critical Analysis

Strengths

• A clear pedagogical route. The publisher describes a gradual path from real analysis to topology with metric spaces as the bridge, and with each concept introduced first in metric spaces and then repeated in topological spaces. The table of contents follows this plan, with three chapters on topological spaces after two on metric spaces.
• Concise and focused. The book has 17 short chapters. In a review for the Mathematical Association of America, Allen Stenger (2010) called it a brief, clearly written introduction to point-set topology.
• Exercises and support material. The publisher notes numerous exercises, with hints for many of the harder ones, and a companion website with further explanations, examples and supplementary material. Rob Ashmore's review in Mathematics Today (December 2011) praised the exercises and the author's teaching style.
• Connects analysis and geometry. One chapter on quotient spaces and surfaces and one on uniform convergence place the abstract theory alongside both the geometric pictures and the analytic results that motivate it.

Limitations

• Introductory depth. Stenger's review says the book introduces the most important concepts of topology but does not take any of them very far. The contents list covers the Hausdorff condition but no wider treatment of separation or countability axioms, and no algebraic topology.
• Answers are not in the book. According to Ashmore, answers to the exercises were dropped from this edition and moved to a website with limited access, which suits lecturers more than students working alone.
• Prerequisite in real analysis. The book is aimed at second- or third-year mathematics students and builds directly on a first analysis course; readers without one should take that course or book first, since Chapter 4 is only a review.
• Page counts differ between catalogues. Library of Congress-based records give 206 pages and Google Books and the IMA listing give 224; check the copy at hand when following page references from a syllabus.

Where It Fits in the Course

Among the topology and geometry books on this site, Sutherland is the bridge from real analysis to point-set topology. Richard Earl's Topology: A Very Short Introduction and Jeffrey Weeks's The Shape of Space are informal and visual, without formal analysis prerequisites, and are good companions for intuition about surfaces. Munkres's Topology is the broader classic that develops point-set topology and then introduces algebraic topology, so it is the natural book to keep for reference or for a second course. Hatcher's Algebraic Topology, free on the author's website, is a graduate text that assumes point-set topology and abstract algebra.

Sutherland sits between these: more formal than the introductions, shorter and more analytic than Munkres, and a route to the point-set background that Hatcher assumes.

How to Use This Book

The publisher aims the book at second- or third-year mathematics students, and the site lists a first real analysis course (sequences, continuity, completeness of the reals) as the prerequisite.

A practical plan:

• Start with Chapters 2 to 4 as a reference. Read them only as far as needed to settle notation and refresh real analysis, then move on.
• Core: Chapters 5 to 14, from metric spaces through topological spaces, subspaces and products, the Hausdorff condition, connectedness and compactness. Treat each metric-space chapter and its topological counterpart as a pair and notice what changes.
• Optional by interest: Chapter 15 for surfaces and gluing, Chapter 16 for uniform convergence, and Chapter 17 for completeness.
• Exercises: use the hints for the harder problems, and check the companion website for further explanations; answers are on a restricted-access site, so work with a study partner or instructor when proofs need checking.