"In geometry, topology is the study of properties of shapes that are independent of size or shape and are not changed by stretching, bending, knotting, or twisting." (M C Escher, 1971)
"[…] under plane transformations, like those encountered in the arbitrary stretching of a rubber sheet, certain properties of the figures involved are preserved. The mathematician has a name for them. They are called continuous transformations. This means that very close lying points pass into close lying points and a line is translated into a line under these transformations. Quite obviously, then, two intersecting lines will continue to intersect under a continuous transformation, and nonintersecting lines will not intersect; also, a figure with a hole cannot translate into a figure without a hole or into one with two holes, for that would require some kind of tearing or gluing - a disruption of the continuity." (Yakov Khurgin, "Did You Say Mathematics?", 1974)
"Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is stretched" in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)
"Geometry is the study of form and shape. Our first encounter with it usually involves such figures as triangles, squares, and circles, or solids such as the cube, the cylinder, and the sphere. These objects all have finite dimensions of length, area, and volume - as do most of the objects around us. At first thought, then, the notion of infinity seems quite removed from ordinary geometry. That this is not so can already be seen from the simplest of all geometric figures - the straight line. A line stretches to infinity in both directions, and we may think of it as a means to go 'far out' in a one-dimensional world." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)
"The problem of coloring a map on the plane is equivalent to the problem on the sphere. If a map on the sphere requires N colors, one can remove one point from the interior of a country on a map on the sphere and stretch the hole open, using stereographic projection, to obtain a map on the plane. The country which had the point removed becomes a sort of ocean surrounding the other countries, and one has a map on the plane using N colors. It should be noted that this projection wildly distorts the relative sizes of countries. [...] Conversely, if all possible maps on the plane require at most N colors, then given any particular map, we can reverse the process above to get a map on a punctured sphere. If the puncture occurs in the middle of a countty, as can be easily arranged, then we obtain a map on the sphere requiring no more than N colors." (L Christine Kinsey. "Topology of Surfaces", 1993)
"Topology deals with those properties of curves, surfaces, and more general aggregates of points that are not changed by continuous stretching, squeezing, or bending. To a topologist, a circle and a square are the same, because either one can easily be bent into the shape of the other. In three dimensions, a circle and a closed curve with an overhand knot in it are topologically different, because no amount of bending, squeezing, or stretching will remove the knot." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Two objects are topologically identical if there is a continuous deformation [...] from one to the other. The bending and stretching [...] are examples of such continuous deformations. Thus, topology is sometimes called the study of continuity, or, a more hackneyed term, 'rubber-sheet geometry', as one pretends everything is formed of extremely flexible rubber." (L Christine Kinsey. "Topology of Surfaces", 1993)
"The best example to indicate the rigidity of analytic functions is a soap film" (with little viscosity) on a wire frame" (think of a bubble blower). The soap film, which is created by the surface tension, stretches across the wire frame and is known to have analyticity. Therefore, if we try to change a certain region of the film by tapping it with a stick, then the film loses analyticity and will immediately brake." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is there any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations." (Marcus du Sautoy, "The Music of the Primes", 1998)
"Topology is the mathematical study of properties of objects which are preserved through deformations, twistings, and stretchings but not through breaks or cuts." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"A continuous function preserves closeness of points. A discontinuous function maps arbitrarily close points to points that are not close. The precise definition of continuity involves the relation of distance between pairs of points. […] continuity, a property of functions that allows stretching, shrinking, and folding, but preserves the closeness relation among points." (Robert Messer & Philip Straffin, "Topology Now!", 2006)
"Topology is geometry without distance or angle. The geometrical objects of study, not rigid but rather made of rubber or elastic, are especially stretchy." (Stephen Huggett & David Jordan, “A Topological Aperitif”, 2009)
"[…] topology is the study of those properties of geometric objects which remain unchanged under bi-uniform and bi-continuous transformations. Such transformations can be thought of as bending, stretching, twisting or compressing or any combination of these." (Lokenath Debnath, "The Legacy of Leonhard Euler - A Tricentennial Tribute", 2010)
"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"At first, topology can seem like an unusually imprecise branch of mathematics. It’s the study of squishy play-dough shapes capable of bending, stretching and compressing without limit. But topologists do have some restrictions: They cannot create or destroy holes within shapes. […] While this might seem like a far cry from the rigors of algebra, a powerful idea called homology helps mathematicians connect these two worlds. […] homology infers an object’s holes from its boundaries, a more precise mathematical concept. To study the holes in an object, mathematicians only need information about its boundaries." (Kelsey Houston-Edwards, "How Mathematicians Use Homology to Make Sense of Topology", Quanta Magazine, 2021)
"In geometry, shapes like circles and polyhedra are rigid objects; the tools of the trade are lengths, angles and areas. But in topology, shapes are flexible things, as if made from rubber. A topologist is free to stretch and twist a shape. Even cutting and gluing are allowed, as long as the cut is precisely reglued. A sphere and a cube are distinct geometric objects, but to a topologist, they’re indistinguishable." (David E Richeson, "Topology 101: The Hole Truth", 2021)
"Topology is the mathematical study of properties of objects which are preserved through deformations, twistings, and stretchings but not through breaks or cuts." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"In geometry, shapes like circles and polyhedra are rigid objects; the tools of the trade are lengths, angles and areas. But in topology, shapes are flexible things, as if made from rubber. A topologist is free to stretch and twist a shape. Even cutting and gluing are allowed, as long as the cut is precisely reglued. A sphere and a cube are distinct geometric objects, but to a topologist, they’re indistinguishable." (David E Richeson, "Topology 101: The Hole Truth", 2021)
"A good analogy is stretching a rubber band. You can stretch and stretch and even feel the tension increase in the muscles in your hands and arms as the gap from one end of the band to the other widens. But at some point you reach the limits of elasticity of the band and it snaps. The same thing happens with human systems." (John L Casti)
"Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything." (Edward Witten)
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