Metric Spaces
by Mícheál Ó Searcóid · Mícheál Ó Searcóid
Springer Undergraduate Mathematics Series textbook building metric space theory slowly from first principles: open and closed sets, convergence, completeness, connectedness, compactness and continuity, with many worked examples. Suits a first course alongside real analysis, preparing readers for general topology.
This link may earn us a small commission at no extra cost to you. Affiliate disclosure
More resources on Metric Spaces
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Real Analysis (MIT 18.100A)
Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs.
Real Analysis (MIT 18.100B)
Proof-based analysis covering real numbers, limits, convergence, series, continuity, metric spaces, differentiation, Riemann integration and differential equations, with emphasis on proving theorems rigorously. Provides 25 lecture videos (about 33 hours), 26 lecture note files, problem sets and exams.
Topology Without Tears
Free textbook by Sidney A. Morris introducing point-set topology from open sets and the euclidean topology through continuity, metric spaces, compactness, connectedness, and Tychonoff's theorem. Readers learn to write rigorous topology proofs, with later chapters extending to topological groups.
Real Mathematical Analysis (2nd edition)
The canonical text learners graduate into. Where most treatments define compactness and move on, Pugh insists on the pictures and then attacks with genuinely difficult exercises, which is what converts recognition into fluency. Springer UTM text from Pugh's honours course at UC Berkeley. Chapter 2, A Taste of Topology, is a long geometric treatment of metric space topology, and Chapter 4 covers function spaces, Arzelà-Ascoli and Stone-Weierstrass. Picture-driven, with famously hard problems.
Metric and Topological Spaces (Cambridge Part IB lecture notes)
Author-hosted Cambridge Mathematical Tripos Part IB notes taking metrics through to abstract topology: open and closed sets, interior and closure, Hausdorff spaces, compactness and products of compact spaces, connectedness, compactness in metric spaces, neighbourhoods. Includes two exercise sets, hints and proofs. Körner argues the motivation for each abstraction rather than just presenting it, and the closing executive summary plus hinted exercises make it usable for self-study rather than only as a lecture crib.