---
title: Topology
description: Topology is the mathematical study of geometric properties preserved under continuous deformations like stretching or bending. Learners will understand concepts of continuity, compactness, and connectedness in abstract spaces.
category: mathematics
subcategory: numerical-analysis-and-topology
difficulty: beginner, intermediate, advanced
url: /subject/topology
---

# Topology

Topology is the mathematical study of geometric properties preserved under continuous deformations like stretching or bending. Learners will understand concepts of continuity, compactness, and connectedness in abstract spaces.

## Available Resources

3 Books • 2 Websites

## Websites

### 1. Topology Without Tears

**Author:** Sidney A. Morris

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.

**Difficulty:** Intermediate | **Price:** Free

**Link:** https://www.topologywithouttears.net/

**Tags:** point-set-topology, general-topology, metric-spaces, compactness, proof-writing

### 2. Hatcher's Algebraic Topology (free PDF)

**Author:** Allen Hatcher

Allen Hatcher's standard graduate textbook, freely downloadable from Cornell, covering fundamental groups, homology, cohomology and homotopy theory, with CW complexes introduced in Chapter 0 and used throughout. Readers learn to build spaces from cells and compute their algebraic invariants.

**Difficulty:** Intermediate | **Price:** Free

**Link:** https://pi.math.cornell.edu/~hatcher/AT/ATpage.html

**Tags:** algebraic-topology, cw-complexes, homology, homotopy-theory, fundamental-group

## Books

### 1. Algebraic Topology

**Author:** Allen Hatcher

Graduate-level algebraic topology covering fundamental groups, homology, cohomology, homotopy theory, and fiber bundles with detailed examples.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://www.math.cornell.edu/~hatcher/AT/AT.pdf

**Tags:** book, mathematics, geometry-&-trigonometry

### 2. Topology - James Munkres

**Author:** James R. Munkres

Standard undergraduate textbook covering point-set topology (open sets, continuity, connectedness, compactness, countability and separation axioms, metrization) followed by an introduction to algebraic topology through the fundamental group and covering spaces. Requires comfort with proofs.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://amzn.to/3NJ7Ob3

**Tags:** point-set-topology, algebraic-topology, fundamental-group, metric-spaces

### 3. Introduction to Topology

**Author:** Bert Mendelson

Short Dover textbook that builds topology from metric spaces to general topological spaces, then treats connectedness and compactness. Includes exercises, and suits readers moving from analysis who want to understand continuity and convergence in abstract settings.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0486663523?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, numerical-analysis-topology

---

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