On Homeomorphism (-1989)

"We consider differential topology to be the study of differentiable manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there exists a differentiable map from one to the other with a differentiable inverse." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"A manifold can be given by specifying the coordinate ranges of an atlas, the images in those coordinate ranges of the overlapping parts of the coordinate domains, and the coordinate transformations for each of those overlapping domains. When a manifold is specified in this way, a rather tricky condition on the specifications is needed to give the Hausdorff property, but otherwise the topology can be defined completely by simply requiring the coordinate maps to be homeomorphisms." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"A diffeomorphism is a differentiable function between manifolds with a differentiable inverse. In particular Φ is one-to-one and onto; thus for example an allocation has a unique supporting price system, and this association is smooth over all optimal allocations." (Steven Smale, "Global Analysis and Economics VI: Geometrical analysis of Pareto Optima and price equilibria under classical hypotheses", 1974)

"Our interest is in pointing out that new surfaces, that is, 2-manifolds, can be formed by fastening together manifolds with boundary along their boundaries, that is, by identifying points of various boundary components by a homeomorphism, assuming of course the necessary condition that such components are homeomorphic." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"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)

"Showing that two spaces are homeomorphic is a geometrical problem, involving the construction of a specific homeomorphism between them. The techniques used vary with the problem. […] Attempting to prove that two spaces are not homeomorphic to one another is a problem of an entirely different nature. We cannot possibly examine each function between the two spaces individually and check that it is not a homeomorphism. Instead we look for 'topological invariants' of spaces: an invariant may be a geometrical property of the space, a number like the Euler number defined for the space, or an algebraic system such as a group or a ring constructed from the space. The important thing is that the invariant be preserved by a homeomorphism- hence its name. If we suspect that two spaces are not homeomorphic, we may be able to confirm our suspicion by computing some suitable invariant and showing that we obtain different answers." (Mark A Armstrong, "Basic Topology", 1979)

"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)

"There are two important groups associated with a closed surface: the fundamental group and the mapping class group. [...] For the fundamental group, choose a base point on the surface and consider the set of all continuous paths that begin and end at this point. The possibility of deforming one such loop into another establishes an equivalence relation which respects the binary operation on loops. [...] For the mapping class group, consider the set of homeomorphisms of the surface to itself (the self-maps). In this context, a self-map is always a oneto-one transformation of a closed, orientable surface onto itself which, together with its inverse, is continuous. The possibility of deforming one self-map into another through a continuous family of self-maps, called an isotopy, establishes an equivalence relation which respects composition of mappings. The product of two self-maps is just the self-map obtained by performing the second immediately after the first." (George K Francis, "A Topological Picturebook", 1987)

"[...] twists about isotopic circles are in the same mapping class. This allows me to shape the pictures conveniently. In describing the effect of a homeomorphism on a surface, it is a good idea to imagine a transparent, flexible copy of the surface lying on top of the fixed copy. The target circle x is on the fixed copy, the working circle y is on the flexible copy. A small isotopy will move y into general position with respect to x, so that y crosses x from one side to the other at a finite number of points. There are global isotopies that reduce such crossings to a minimum, even among an entire generating set of circles [...]"(George K Francis, "A Topological Picturebook", 1987)

"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)