"A homeomorphism may be thought of as the best possible type of continuous function, and homeomorphic spaces are considered the same in topology. [...] A complex has two structures: that of the topological space underlying the complex, and the subdivision of the complex into cells. The corresponding functions will have to preserve this dual nature. In particular, it would be nice if these functions, yet to be defined, would also induce nice functions on the homology of the complex. Since homology has an algebraic group structure, we want the functions to induce homomorphisms on the homology groups. The best of all functions would be homeomorphisms on the underlying topological space and also isomorphisms oii the homology groups." (L Christine Kinsey. "Topology of Surfaces", 1993)
"Compactness is an important property and is the topologists' form of finiteness, in the sense that a compact set does not go on forever. One's first impression is that a bounded set should be 'finite', but we have seen that the interval ( -1, 1) is topologically equivalent to ( -∞, ∞). One way of thinking about this is to imagine walking along the interval ( -1, 1), towards 1, but for some reason (an increase in gravity, or the unnerving fact that one's legs seem to be getting shorter and shorter), the closer one gets to 1, the smaller steps one has to take, so that one never reaches 1. In this sense, ( -1, 1) is endless. On the other hand, [-1, 1] is compact and does not go on forever, since it has ends. No topological property should be based solely on distance, as is boundedness, since in topology distance means very little." (L Christine Kinsey. "Topology of Surfaces", 1993)
"In any mathematical study, the first thing to do is specify the type or category of objects to be investigated. In topology, the most general possible objects of study are sets of points with just enough structure to be able to define continuous functions." (L Christine Kinsey. "Topology of Surfaces", 1993)
"In most fields of mathematics, we study some class of objects and also the functions appropriate to the objects of study. In linear algebra, the objects studied are vector spaces and the corresponding functions, called linear transformations and usually represented by matrices, are those which preserve the vector space structure. In geometry, we study isometries, such as rotations: functions that do not change the geometric properties of length, angle measure, area, and volume. In abstract algebra, the objects of study are groups and the appropriate functions are homomorphisms, which preserve the algebraic properties of the groups. The best possible homomorphism is an isomorphism; isomorphic groups are essentially identical algebraically." (L Christine Kinsey. "Topology of Surfaces", 1993)
"It would be preferable to have some property or quantity that could indisputably differentiate between two surfaces. This, however, requires more sophisticated mathematics. [...] we introduce the simplest topological invariant: the euler characteristic. This is the first step in the algebraization of topology: finding something computable to describe the shape of a space." (L Christine Kinsey. "Topology of Surfaces", 1993)
"Intrinsic properties have to do with the object itself, in contrast to extrinsic properties which describe how the object is embedded in the surrounding space. The cylinder and the band with two twists are intrinsically the same: There is a homeomorphism between them. The difference between them lies in how they sit in our 3-dimensional universe. Both are assembled from a rectangle by the same gluing instructions but one is given two twists before gluing." (L Christine Kinsey. "Topology of Surfaces", 1993)
"The Möbius band has the interesting property of having only one side, in contrast to the cylinder. It is easy to imagine a cylinder with the outside painted one color and the inside another. Try painting a Mobius strip. Another peculiarity of the Mobius band occurs when one cuts it along the dotted line (called the meridian), and then follows the gluing instructions of the edges [...]. The end result is a planar diagram for a cylinder. However, if one actually constructs a Möbius band of paper and cuts it as described above, one gets something that looks like a cylinder with two twists." (L Christine Kinsey. "Topology of Surfaces", 1993)
"Topology is a relatively new field of mathematics and is related to geometry. In both of these subjects one studies the shape of things. In geometry, one characterizes, for example, a can of pineapple by its height, radius, surface area, and volume. In topology, one tries to identify the more subtle property that makes it impossible to get the pineapple out of the tin, no matter what shape it is battered into, as long as one does not puncture the can." (L Christine Kinsey. "Topology of Surfaces", 1993)
"The Euler characteristic can be used to shorten the process, but for some cases a lengthy procedure is still necessary. Neither of these options provide a clear accounting for the ways in which the surfaces vary, e.g., which enclose cavities, which are non-orientable, etc., and neither can be completely generalized to higher-dimensional manifolds. Ideally, one would like some sort of algebraic invariant or computable quantity that would codify a lot of information: how many connected pieces a space has, how the gluing directions work, whether the surface is orientable or not, etc. The euler characteristic is a first attempt at this and has the advantage of being quite easy to compute, but it fails to distinguish between the torus and the Klein bottle, which both have x = 0." (L Christine Kinsey. "Topology of Surfaces", 1993)
"The euler characteristic is strikingly easy to compute, and it really seems odd that it should be a topological invariant. After all, it is computed from a cell decomposition, and it is very easy to come up with a lot of different complexes on a single surface, all with varying numbers of faces, edges, and vertices. But somehow by taking the alternating sum, one arrives at a quantity which depends only on the underlying shape and not on the particular complex." (L Christine Kinsey. "Topology of Surfaces", 1993)
"The object of topology is the classification and description of the shape of a space up to topological equivalence." (L Christine Kinsey. "Topology of Surfaces", 1993)
"The problem of coloring a map on the plane is equivalent to the problem on the sphere. If a map on the sphere requires N colors, one can remove one point from the interior of a country on a map on the sphere and stretch the hole open, using stereographic projection, to obtain a map on the plane. The country which had the point removed becomes a sort of ocean surrounding the other countries, and one has a map on the plane using N colors. It should be noted that this projection wildly distorts the relative sizes of countries. [...] Conversely, if all possible maps on the plane require at most N colors, then given any particular map, we can reverse the process above to get a map on a punctured sphere. If the puncture occurs in the middle of a countty, as can be easily arranged, then we obtain a map on the sphere requiring no more than N colors." (L Christine Kinsey. "Topology of Surfaces", 1993)
"Two objects are topologically identical if there is a continuous deformation [...] from one to the other. The bending and stretching [...] are examples of such continuous deformations. Thus, topology is sometimes called the study of continuity, or, a more hackneyed term, 'rubber-sheet geometry', as one pretends everything is formed of extremely flexible rubber." (L Christine Kinsey. "Topology of Surfaces", 1993)
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