On Topology VIII

"Topology has to do with those properties of a space which are left unchanged by the kind of transformation that we have called a topological equivalence or homeomorphism. But what sort of spaces interest us and what exactly do we mean by a 'space? The idea of a homeomorphism involves very strongly the notion of continuity [...]"  (Mark A Armstrong, "Basic Topology", 1979)

"Fractal geometry is concerned with the description, classification, analysis, and observation of subsets of metric spaces (X, d). The metric spaces are usually, but not always, of an inherently 'simple' geometrical character; the subsets are typically geometrically 'complicated'. There are a number of general properties of subsets of metric spaces, which occur over and over again, which are very basic, and which form part of the vocabulary for describing fractal sets and other subsets of metric spaces. Some of these properties, such as openness and closedness, which we are going to introduce, are of a topological character. That is to say, they are invariant under homeomorphism." (Michael Barnsley, "Fractals Everwhere", 1988)

"[…] it is useful to note that there are three basic network topologies. First, there are line or chain networks with many nodes that are spread out in more or less linear fashion. Second, there are star or hub networks, where most important relationships move through a central hub or hubs. Third, there are all-channel networks, in which communications proceed in more or less all directions across the network simultaneously […]." (John Urry, "Global Complexity", 2003)

"[…] topology, the study of continuous shape, a kind of generalized geometry where rigidity is replaced by elasticity. It's as if everything is made of rubber. Shapes can be continuously deformed, bent, or twisted, but not cut - that's never allowed. A square is topologically equivalent to a circle, because you can round off the corners. On the other hand, a circle is different from a figure eight, because there's no way to get rid of the crossing point without resorting to scissors. In that sense, topology is ideal for sorting shapes into broad classes, based on their pure connectivity." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Topology is the study of geometric objects as they are transformed by continuous deformations. To a topologist the general shape of the objects is of more importance than distance, size, or angle." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"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)

"Topology is a child of twentieth century mathematical thinking. It allows us to consider the shape and structure of an object without being wedded to its size or to the distances between its component parts. Knot theory, homotopy theory, homology theory, and shape theory are all part of basic topology. It is often quipped that a topologist does not know the difference between his coffee cup and his donut - because each has the same abstract 'shape' without looking at all alike." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"A topological property is, therefore, any property that is preserved under the set of all homeomorphisms. […] Homeomorphisms generally fail to preserve distances between points, and they may even fail to preserve shapes." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)