On Topology III (Topology with a Twist)

"The connection of topology with physics is no passing interlude but rather represents a length affair." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language, whose beauty eventually - in retrospect - compensates for all the drudgery." (Volker Runde, "A Taste of Topology", 2005)

"[…] geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset." (Henri Poincaré, 1895)

"People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook in topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram." (Martin Gardner, "Hexaflexagons and Other Mathematical Diversions", 1988)

"A child[’s …] first geometrical discoveries are topological…If you ask him to copy a square or a triangle, he draws a closed circle." (Jean Piaget)

"If you wear glasses, and you wake up in the morning and you’re not wearing your glasses, and everything is blurred together, that’s what the indiscrete topology is like." (Anonymous)

"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl)

"Point set topology is a disease from which the human race will soon recover." (Henri Poincaré)

"The true traditional doughnut has the topology of a sphere. It is a matter of taste whether one regards this as having separate internal and external surfaces. The important point is that the inner space should be filled with good raspberry jam. This is also a matter of taste." (Peter B Fellgett)

"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)