"The problem of determining the global structure of a space form from its local metric properties and the connected one of metrizing - in the sense of differential geometry - a given topological space, may be worthy of interest for physical reasons." (Heinz Hopf, 1932)
"Speaking roughly, a homology theory assigns groups to topological spaces and homomorphisms to continuous maps of one space into another. To each array of spaces and maps is assigned an array of groups and homomorphisms. In this way, a homology theory is an algebraic image of topology. The domain of a homology theory is the topologist’s field of study. Its range is the field of study of the algebraist. Topological problems are converted into algebraic problems." (Samuel Eilenberg & Norman E Steenrod, "Foundations of Algebraic Topology", 1952)
"In all candor, we must admit that the intuitive meaning of compactness for topological spaces is somewhat elusive. This concept, however, is so vitally important throughout topology […]" (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)
"A surface is a topological space in which each point has a neighbourhood homeomorphic to the plane, ad for which any two distinct points possess disjoint neighbourhoods. […] The requirement that each point of the space should have a neighbourhood which is homeomorphic to the plane fits exactly our intuitive idea of what a surface should be. If we stand in it at some point (imagining a giant version of the surface in question) and look at the points very close to our feet we should be able to imagine that we are standing on a plane. The surface of the earth is a good example. Unless you belong to the Flat Earth Society you believe it to,be (topologically) a sphere, yet locally it looks distinctly planar. Think more carefully about this requirement: we ask that some neighbourhood of each point of our space be homeomorphic to the plane. We have then to treat this neighbourhood as a topological space in its own right. But this presents no difficulty; the neighbourhood is after all a subset of the given space and we can therefore supply it with the subspace topology." (Mark A Armstrong, "Basic Topology", 1979)
"[…] in trying to prove a concrete geometrical result such as the classification theorem for surfaces, the purely topological structure of the surface (that it be locally euclidean) does not give us much leverage from which to start. On the other hand, although we can define algebraic invariants, such as the fundamental group, for topological spaces in general, they are not a great deal of use to us unless we can calculate them for a reasonably large collection of spaces. Both of these problems may be dealt with effectively by working with spaces that can be broken up into pieces which we can recognize, and which fit together nicely, the so called triangulable spaces." (Mark A Armstrong, "Basic Topology", 1979)
"Since geometry is the mathematical idealization of space, a natural way to organize its study is by dimension. First we have points, objects of dimension O. Then come lines and curves, which are one-dimensional objects, followed by two-dimensional surfaces, and so on. A collection of such objects from a given dimension forms what mathematicians call a 'space'. And if there is some notion enabling us to say when two objects are 'nearby' in such a space, then it's called a topological space." (John L Casti, "Five Golden Rules", 1995)
"[...] if we consider a topological space instead of a plane, then the question of whether the coordinates axes in that space are curved or straight becomes meaningless. The way we choose coordinate systems is related to the way we observe the property of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"One of the basic tasks of topology is to learn to distinguish nonhomeomorphic figures. To this end one introduces the class of invariant quantities that do not change under homeomorphic transformations of a given figure. The study of the invariance of topological spaces is connected with the solution of a whole series of complex questions: Can one describe a class of invariants of a given manifold? Is there a set of integral invariants that fully characterizes the topological type of a manifold? and so forth." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
"The primary goal in general topology, also sometimes called point set topology, is the investigation and comparison of different classes of topological spaces. This primary goal continues to yield interesting problems and results, which derive their significance from their relevance with respect to this primary goal and from the need of applications." (Teun Koetsier & Jan van Mill, "By Their Fruits Ye Shall Know Them: Some Remarks on the Interaction of General Topology with Other Areas of Mathematics", [in Ioan M. James, "History of Topology"], 1999)
"To describe the property of smoothness, differentiable functions should be specified first. To do so, coordinates need to be introduced on the topological space. Those coordinates can be local coordinates such as the ones used by Gauss. Once coordinates are introduced around a point a in a topological space, differentiable functions near the point a are distinguished from the continuous functions in the region near a. If different coordinates are chosen, then a different set of differentiable functions is distinguished. In other words, the choice of local coordinates determines the notion of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"The most fundamental tool in the subject of point-set topology is the homeomorphism. This is the device by means of which we measure the equivalence of topological spaces." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)
"In each branch of mathematics it is essential to recognize when two structures are equivalent. For example two sets are equivalent, as far as set theory is concerned, if there exists a bijective function which maps one set onto the other. Two groups are equivalent, known as isomorphic, if there exists a a homomorphism of one to the other which is one-to-one and onto. Two topological spaces are equivalent, known as homeomorphic, if there exists a homeomorphism of one onto the other." (Sydney A Morris, "Topology without Tears", 2011)
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