6 quotes found
"Milnor's arguments were breathtaking. He brought together topology and analysis in a wholly unexpected way, and in doing so initiated the field of differential topology."
"Presentday topology consists of two distinct parts: point set topology and algebraic topology. The first has mainly been the prerogative of Poland plus a strong American component: the school of R. L. Moore (of Austin, Texas)."
"If geometry is dressed in a suit coat, topology dons jeans and a T-shirt."
"Topologists are interested not only in finite-dimensional spaces (for example, subspaces of Rn), but also in infinite-dimensional ones, such as the spaces occurring in quantum field theory."
"In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain."
"For a normal topological space, there exist three different approaches to define its dimension, due to Lebesgue and Čech (covering dimension dim), Urysohn and Menger (small inductive dimension ind), and Poincaré, Brouwer, and Čech (large inductive dimension Ind). All the three dimensions coincide for separable metric spaces (Tumarkin-Urysohn–Hurewitz's theorem proved in the late 20s), the equality dim X = Ind X holds for any metric space X (Katetov–Morita's theorem, 1954), while in 1962, Roy [Roy] constructed his famous example of a complete metric space Y such that ind Y = 0, but dim Y = Ind Y = 1."