topology: Quick Facts

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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.

A torus is shown with three labeled loops: one encircling the central hole (a), one passing through the hole (b), and one small loop on the surface (c), illustrating concepts in general 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.

Diagram illustrating a topological mapping from set X to set Y, with a point p in subset U of X mapped to f(p) in subset f(U) of Y, showing the concept of continuity in topology.

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.

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Riemann surface differential topology point-set topology general topology hypercube knot theory topological space Jordan curve theorem Möbius strip catastrophe theory analysis probability theory cryptology computer science combinatorics algebra statistics automata theory set theory information theory

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Henri Poincaré

French mathematician

Pavel Sergeevich Aleksandrov

Soviet mathematician

Luitzen Egbertus Jan Brouwer

Dutch mathematician

August Ferdinand Möbius

German mathematician and astronomer

$Sierpiński gasketPolish mathematician Wacław Sierpiński described the fractal that bears his name in 1915, although the design as an art motif dates at least to 13th-century Italy. Begin with a solid equilateral triangle, and remove the triangle formed by connecting the midpoints of each side. The midpoints of the sides of the resulting three internal triangles are connected to form three new triangles that are then removed to form nine smaller internal triangles. The process of cutting away triangular pieces continues indefinitely, producing a region with a Hausdorf dimension of a bit more than 1.5 (indicating that it is more than a one-dimensional figure but less than a two-dimensional figure).$

Wacław Sierpiński

Polish mathematician

Sir Michael Francis Atiyah

British mathematician

Alexandre Grothendieck

German-French mathematician

Simon Donaldson

British mathematician

Saunders Mac Lane

American mathematician

Michael Freedman

American mathematician

Enrico Betti

Italian mathematician

Lev Semyonovich Pontryagin

Russian mathematician

James W. Alexander II

American mathematician

Henry Whitehead

British mathematician

Max Dehn

German mathematician

Oswald Veblen

American mathematician

Stephen Smale

American mathematician

David Thouless

British-born American physicist

Sergei Novikov

Russian mathematician

William Paul Thurston

American mathematician

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