Topology 3
Topology 3 : A branch of geometry concerned with the general properties of shapes and space. It can be thought of as the study of properties that are not changed by continuous deformations, such as stretching or twisting. A sphere and an ellipsoid are different figures in solid (Euclidean) geometry, but in topology they are considered equivalent since one can be transformed into the other by a continuous deformation. A torus, on the other hand, is not topologically equivalent to a sphere - it would not be possible to distort a sphere into a torus without breaking or joining surfaces. A torus is thus a different type of shape to a sphere. Topology studies types of shapes and their properties. Topology uses methods of higher algebra including group theory and set theory. An important notion is that of sets of points in the neighbourhood of a given point (i.e. within a certain distance of the point). An open set is a set of points such that each point in the set has a neighbourhood containing points in the set. A topological transformation occurs when there is a one-to-one correspondence between points in one figure and points in another so that open sets in one correspond to open sets in the other. If one figure can be transformed into another by such a transformation, the sets are topologically equivalent
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