topology

Table of Contents

References & Edit History Quick Facts & Related Topics

Images & Videos

Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

The torus is not simply connected. While the small loop c can be shrunk to a point without breaking the loop or the torus, loops a and b cannot because they encompass the torus's central hole.

Understand the hairy ball theorem topology

In knot theory, knots are formed by seamlessly merging the ends of a segment to form a closed loop. Knots are then characterized by the number of times and the manner in which the segment crosses itself. After the basic loop, the simplest knot is the trefoil knot, which is the only knot, other than its mirror image, that can be formed with exactly three crossings.

The topological concept of a continuous functionA function f from a topological space X to a topological space Y is continuous at p ∊ X if, for any neighbourhood V of f(p), there exists a neighbourhood U of p such that f(U) ⊆ V.

Mathematical knotsKnots are characterized by the number of times and the manner in which the strand crosses itself. A basic loop, which has no crossings and forms only one distinct “knot,” is the simplest knot in knot theory. The number of distinct knots greatly increases with the number of crossings; only those with seven or fewer crossings are shown here.

For Students

topology summary

Discover

solar eclipse, sun, moon, astronomy, space

How Do You Tell the Difference Between Total, Annular, Solar, and Lunar Eclipses?

9 of the World’s Deadliest Spiders

A Factory Interior, watercolor, pen and gray ink, graphite, and white goache on wove paper by unknown artist, c. 1871-91; in the Yale Center for British Art. Industrial Revolution England

Inventors and Inventions of the Industrial Revolution

Aphrodite. Greek mythology. Sculpture. Aphrodite is the Greek goddess of love and beauty.

12 Greek Gods and Goddesses

The Sun Was Eaten: 6 Ways Cultures Have Explained Eclipses

Julius Caesar in full Gaius Julius Caesar (100? BCE-44 BCE) statue in Rimini, Italy. Roman general and statesman and dictator

The Time Julius Caesar Was Captured by Pirates

VAN, TURKEY - OCTOBER 25: A building and car ruined during the earthquake of Van-Ercis on October 25, 2011 in Van, Turkey. It is 604 killed and 4152 injured in Van-Ercis Earthquake.

What Is an Aftershock?

Science & Tech

topology

Also known as: analysis situs

Written by Stephan C. Carlson

Fact-checked by The Editors of Encyclopaedia Britannica

Last Updated: Feb 28, 2024 • Article History

topology

See all media

Category: Science & Tech

Key People:: Henri Poincaré; Pavel Sergeevich Aleksandrov; Luitzen Egbertus Jan Brouwer; August Ferdinand Möbius; Sir Michael Francis Atiyah

Related Topics:: knot theory; topological space; Jordan curve theorem; catastrophe theory; Klein bottle

See all related content →

topology, branch of mathematics, sometimes referred to as “rubber sheet geometry,” in which two objects are considered equivalent if they can be continuously deformed into one another through such motions in space as bending, twisting, stretching, and shrinking while disallowing tearing apart or gluing together parts. The main topics of interest in topology are the properties that remain unchanged by such continuous deformations. Topology, while similar to geometry, differs from geometry in that geometrically equivalent objects often share numerically measured quantities, such as lengths or angles, while topologically equivalent objects resemble each other in a more qualitative sense. The area ...(100 of 2963 words)

Access the full article

Help support true facts by becoming a member.