Topology supplies a language in which continuity, compactness and connectedness survive without a distance formula. Morris develops that language through definitions, examples and exercises. Hatcher provides a shorter bridge toward algebraic topology. Use the longer route when you need practice recognizing topologies and continuous maps; use the shorter account when familiar metric-space intuition needs translation into open-set arguments.
What to work on
Prove continuity using inverse images of open sets.
Distinguish connectedness, compactness and separation assumptions in examples.
Use bases, subspaces and product topologies.
Verify the defining property of a quotient topology.
A starting text
Topology Without Tears — Sidney A. Morris. Read 1.1-1.3 on topologies, 2.2-2.3 on bases, 3.1 and 3.3 on closure and connectedness, 4.1 and 5.1 on subspaces and continuity, 7.1-7.2 on compactness, 8.1-8.2 on products, and 11.1 on quotient spaces.
Choosing what to read
Test every definition on a finite space, a metric space and a space with a less familiar topology. Prove continuity once by inverse images of opens and once in a metric example, explaining why sequences are not a universal replacement. For compactness, work directly with an open cover and a finite subcover. When constructing a quotient, describe the quotient map and the topology it induces before drawing a picture. Keep counterexamples close to the separation assumptions they test.
Practice open-set reasoning, compactness, products and quotients.
How it reads
Detailed introductory route with many definition-testing exercises.
What it expects
Proofs, sets and elementary continuity examples.
What to keep in mind
The author explicitly does not provide exercise solutions; worked examples are a different form of support.
Practice
Exercises and worked examples are public. On the title page the author explicitly states that he does not provide solutions to the exercises. Exercise sets 3.3, 5.1, 7.1-7.2, 8.1 and 11.1. In Exercises 11.1, questions 1-4 check the quotient-map definition and composition.
Complete public PDF.
Edition, reading range and source
Read 1.1-1.3 on topologies, 2.2-2.3 on bases, 3.1 and 3.3 on closure and connectedness, 4.1 and 5.1 on subspaces and continuity, 7.1-7.2 on compactness, 8.1-8.2 on products, and 11.1 on quotient spaces.
Topology exercises and worked examples. Exercise sets 3.3, 5.1, 7.1-7.2, 8.1 and 11.1. In Exercises 11.1, questions 1-4 check the quotient-map definition and composition.. Exercises and worked examples are public. On the title page the author explicitly states that he does not provide solutions to the exercises.
Try these before moving on
Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.
Prove that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Give a continuous bijection that fails to be a homeomorphism when these hypotheses are dropped.
A hint
Show the bijection is closed. For a counterexample, compare two topologies on the same two-point set.
Work through the solution
Let f:X→Y be a continuous bijection, with X compact and Y Hausdorff. A closed subset C⊆X is compact: append X∖C to any open cover of C and use compactness of X. Its image f(C) is compact by continuity. A compact subset K of a Hausdorff space is closed: for y∈/K, separate y from each k∈K by disjoint open sets, cover K by finitely many of the neighborhoods of k, and intersect the corresponding neighborhoods of y. This produces an open neighborhood of y disjoint from K. Thus f is closed. For g=f−1, the preimage g−1(C)=f(C) of every closed C⊆X is closed in Y, proving continuity of g.
For a counterexample, let X={0,1} be discrete and let the same underlying set Y have the indiscrete topology {∅,Y}. The identity map X→Y is a continuous bijection. Its inverse is not continuous, because the inverse image of the open singleton {0}⊆X is not open in Y. Here X is compact, but Y is not Hausdorff, so the theorem's hypotheses are not all present.
Set Theory — Review the difference between a set bijection and additional structure on the sets.
Already comfortable with this? Skip the definitions you can use correctly in proofs. Skip this selection only if both checks, a product-topology argument, and the distinction between connectedness and path connectedness are secure.
A first course emphasizing topological definitions and proof practice. Useful before algebraic topology; elementary prerequisites do not mean the abstractions require little effort.
Preparation: Sets, functions and proof techniques.
Oleg Viro, Oleg Ivanov, Nikita Netsvetaev and Viatcheslav Kharlamov · Problem textbook · Introductory · Russian · PDF
The author’s university book page links the complete Russian PDF.
A problem-led development of general topology, followed by fundamental groups and covering spaces. This record uses the full Russian book, not the shortened English file without proofs.
Preparation: Set theory; proof writing; elementary group theory for later sections.
Use: Work through examples and proof exercises before a homology course.
Anton Petrunin · Course notes · Introductory · English · PDF
The complete PDF opens on the author’s or university’s public site.
An exercise-driven course from metric and topological spaces through compactness, connectedness, fundamental groups and coverings; the author labels the notes incomplete.
Preparation: Elementary sets; calculus; comfort with proofs.
Use: Use as a first topology course and bridge to algebraic topology.
Lynn Arthur Steen and J. Arthur Seebach Jr. · Textbook · Intermediate · English · Purchase / institutional access
Purchase or institutional access required; previews are not the complete book.
A reference of examples distinguishing separation, compactness, connectedness and related properties. Use it alongside a systematic course rather than as its replacement.
Preparation: Proof-writing, sets and elementary real analysis.
Klaus Jänich · Textbook · Intermediate · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
A concise general-topology course with geometric examples. Use it to reinforce continuity, compactness and product constructions before manifold topology.
Satish Shirali and Harkrishan Lal Vasudeva · Textbook · Intermediate · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
Develops metric-space techniques linking real analysis and topology. Useful for practicing completeness, compactness and continuity with concrete examples.
Christian Berg; 1997 revision by Bergfinnur Durhuus · Course notes · Intermediate · Danish · PDF
University department archive supplies the complete 96-page course book, including the revision preface and exercises.
English title: Metric Spaces. Connects distance and continuity with completeness, compactness and the contraction theorem, ending with differential-equation applications.
Preparation: Single-variable analysis, basic linear algebra and elementary proofs.
Use: Bridge concrete limits to metric-space arguments through the accompanying exercises.
方恪身 (Kesheng Fang) · Course notes · Intermediate · Chinese · Online text
Online lecture text; chapters can be read separately.在线讲义，可按章节阅读。
A course connecting point-set topology with fundamental groups and covering spaces. The introduction states that this 2021 lecture series is no longer being updated.从点集拓扑进入基本群与覆盖空间的课程讲义。前言注明这是2021年的课程材料，已经停止更新。
Preparation: Sets, maps and elementary group theory.集合、映射和初等群论。
Use: Follow the topology chapters before the fundamental-group and covering-space material.先读拓扑空间部分，再读基本群与覆盖空间。
John R. Stallings · Long lecture notes · Graduate · English · PDF
Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.
A bridge from general topology into polyhedra and geometric topology. Use after basic compactness and quotient-space constructions.
Stephen Willard · Textbook · Graduate · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
An extensive general-topology reference for separation, compactness and convergence. Compare examples with an introductory course before using it as a reference.