On Homeomorphism I

"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 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)

"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Homeomorphism is one of the basic concepts in topology. Homeomorphism, along with the whole topology, is in a sense the basis of spatial perception. When we look at an object, we see, say, a telephone receiver or a ring-shaped roll and first of all pay attention to the geometrical shape (although we do not concentrate on it specially) - an oblong figure thickened at the ends or a round rim with a large hole in the middle. Even if we deliberately concentrate on the shape of the object and forget about its practical application, we do not yet 'see' the essence of the shape. The point is that oblongness, roundness, etc. are metric properties of the object. The topology of the form lies 'beyond them'." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"The concept of homeomorphism appears to be convenient for establishing those important properties of figures which remain unchanged under such deformations. These properties are sometimes referred to as topological, as distinguished from metrical, which are customarily associated with distances between points, angles between lines, edges of a figure, etc." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

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

"Note that a stable homeomorphism must preserve orientation, if X is an orientable manifold. Note also that it is easier to observe this fact than to define the term orientation." (Edwin E Moise, "Isotopies", 1997)