On Topology II (Definitions)

"I believe that we need another analysis properly geometric or linear, which treats PLACE directly the way that algebra treats MAGNITUDE." (Gottfried W Leibniz, 1670s)

"The branch of geometry that deals with magnitudes has been zealously studied throughout the past, but there is another branch that has been almost unknown up to now; Leibniz spoke of it first, calling it the ‘geometry of position’ (geometria situs). This branch of geometry deals with relations dependent on position; it does not take magnitudes into considerations, nor does it involve calculation with quantities. But as yet no satisfactory definition has been given of the problems that belong to this geometry of position or of the method to be used in solving them." (Leonhard Euler, 1735) 

"Topology is the study of the modal relations of spatial figures and the laws of connectivity, mutual position, and ordering of points, lines, surfaces, and solids and their parts independently of measure and magnitude relations." (Johann B Listing)

"The use of figures is, above all, then, for the purpose of making known certain relations between the objects that we study, and these relations are those which occupy the branch of geometry that we have called Analysis Situs [that is, topology], and which describes the relative situation of points and lines on surfaces, without consideration of their magnitude." (Henri Poincaré, "Analysis Situs", Journal de l'Ecole Polytechnique 1, 1895)

"Imagine any sort of model and a copy of it done by an awkward artist: the proportions are altered, lines drawn by a trembling hand are subject to excessive deviation and go off in unexpected directions. From the point of view of metric or even projective geometry these figures are not equivalent, but they appear as such from the point of view of geometry of position [that is, topology]." (Henri Poincaré, "Dernières pensées", 1920)

"Topology begins where sets are implemented with some cohesive properties enabling one to define continuity." (Solomon Lefschetz, "Introduction to Topology", 1949)

"In topology we are concerned with geometrical facts that do not even involve the concepts of a straight line or plane but only the continuous connectiveness between points of a figure." (David Hilbert, "Geometry and Imagination", 1952)

"Topology is precisely that mathematical discipline which allows a passage from the local to the global." (René Thom)

"Topology studies the properties of geometrical objects that remain unchanged under transformations called homeomorphisms and deformations." (Victor V Prasolov, "Intuitive Topology", 1995)

"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)

"Topology makes it possible to explain the general structure of the set of solutions without even knowing their analytic expression." (Michael I Monastyrsky, "Riemann, Topology, and Physics" 2nd Ed., 2008)

"[…] 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)