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topology
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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. 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.
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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.
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.
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.
Understand the hairy ball theorem topology
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Key People

Luitzen Egbertus Jan Brouwer
Dutch mathematician
Henri Poincaré, 1909.
Henri Poincaré
French mathematician
Aleksandrov, Pavel Sergeevich
Pavel Sergeevich Aleksandrov
Soviet mathematician
August Ferdinand Möbius, detail from an engraving by an unknown artist.
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, 2004.
Sir Michael Francis Atiyah
British mathematician
Alexandre Grothendieck
German-French mathematician
Simon Donaldson
British mathematician
Saunders Mac Lane
American mathematician
Michael Freedman
Michael Freedman
American mathematician
Enrico Betti
Italian mathematician
Pontryagin, Lev Semyonovich
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
Thouless, David
David Thouless
British-born American physicist
Sergei Novikov
Russian mathematician
William Paul Thurston
American mathematician

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