Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.
Möbius published a description of a Möbius band in 1865. He tried to describe the 'one-sided' property of the Möbius band in terms of non-orientability. He thought of the surface being covered by oriented triangles. He found that the Möbius band could not be filled with compatibly oriented triangles.
The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180 degrees, and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation.
Königsberg bridge problem, a recreational mathematical puzzle, set in the old Prussian city of Königsberg (now Kaliningrad, Russia), that led to the development of the branches of mathematics known as topology and graph theory. In the early 18th century, the citizens of Königsberg spent their days walking on the intricate arrangement of bridges across the waters of the Pregel (Pregolya) River, which surrounded two central landmasses connected by a bridge (3). Additionally, the first landmass (an island) was connected by two bridges (5 and 6) to the lower bank of the Pregel and also by two bridges (1 and 2) to the upper bank, while the other landmass (which split the Pregel into two branches) was connected to the lower bank by one bridge (7) and to the upper bank by one bridge (4), for a total of seven bridges. According to folklore, the question arose of whether a citizen could take a walk through the town in such a way that each bridge would be crossed exactly once.
The motions associated with a continuous deformation from one object to another occur in the context of some surrounding space, called the ambient space of the deformation. When a continuous deformation from one object to another can be performed in a particular ambient space, the two objects are said to be isotopic with respect to that space.
Topology is the study of sets on which one has a notion of "closeness" -- enough to decide which functions defined on it are continuous. Thus it is a kind of generalized geometry (we are still interested in spheres and cubes, for example, but we might consider them to be "the same", yet distinct from a bicycle tire, which has a "hole") or a kind of generalized analysis (we might think of the functions f(x)=x^2 and f(x)=|x| as being "the same", and yet distinct from f(x)=signum(x)=x/|x|, which has a discontinuity).
After all, in dimensions zero, one, and two, there is not much that can happen, and besides, we as three-dimensional creatures can visualize much of it easily. You might think that dimension three would be fine, too, but remember, the kind of dimension we are discussing is intrinsic dimension. To visualize it we would have to live in at least four dimensions. It turns out, however, that much of this visualization is irrelevant in the final analysis anyway, since you still need to mathematically prove your results, which is more demanding than simply drawing a picture and staring at it. But at the very least, the manifolds can become more and more strange as you increase in dimension. So the higher the dimension, the more difficult the situation might be.
In two dimensions, a triangle, a square, and a circle are all topologiclaly equivalent. So are the upper case letters "T", "F" and "E". In 3 dimensions, the surface of a cube, a pyramid, and a sphere are topologically equivalent. A stretchable "skin" that covers any one of them can be restretched to cover any of the others. The surface of a donut and a coffee cup are topologically equivalent--each is a three-dimensional object with a hole in it. This is an unusual (but valid) way to think about the world.
Topology is almost the most basic form of geometry there is. It is used in nearly all branches of mathematics in one form or another. There is an even more basic form of geometry called homotopy theory, which is what I actually study most of the time. We use topology to describe homotopy, but in homotopy theory we allow so many different transformations that the result is more like algebra than like topology. This turns out to be convenient though, because once it is a kind of algebra, you can do calculations, and really sort things out! And, surprisingly, many things depend only on this more basic structure (homotopy type), rather than on the topological type of the space, so the calculations turn out to be quite useful in solving problems in geometry of many sorts.
Topology began with the study of curves, surfaces, and other objects in the plane and three-space. One of the central ideas in topology is that spatial objects like circles and spheres can be treated as objects in their own right, and knowledge of objects is independent of how they are "represented" or "embedded" in space. For example, the statement "if you remove a point from a circle, you get a line segment" applies just as well to the circle as to an ellipse, and even to tangled or knotted circles, since the statement involves only topological properties.
Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus (i.e., a two-dimensional a surface that can be embedded in three-dimensional space), and the set of all possible positions of the hour, minute, and second hands taken together are topologically equivalent to a three-dimensional object.