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Introduction of Topology

Topology is the study of connectivity and continuity. In another word topology is a category of mathematics which can be formally explained as "the study of qualitative properties of certain objects (known as topological spaces) which are invariant under a certain type of transformation particularly those properties which are invariant under a certain type of equivalence, The word or term "topology" is sometimes used to define a structure forced upon a set S, a structure which necessarily characterizes the set S as a topological space by taking proper care of properties like connectedness, convergence, and continuity, upon transformation. Topological spaces explain automatically in almost every branch of mathematics. This made topology one of the big unifying ideas of mathematics. The motivating insight behind the topology is that some geometric problems based not on the exact shape of the objects involved but instead on the way they are put altogether. For illustration, the square and the circle have many properties in general: they are both one dimensional object and both divide the plane into two parts which is inside and outside. Topology is a main region of mathematics concerned with the most fundamental properties of space, like connectedness. More accurately, topology studies properties which are preserved under continuous deformations, involving bending and stretching, but not gluing or tearing. Topology developed as a field of study out of set theory and geometry, throughout the analysis of such notions as transformation, space, and dimension.

The topology has many sub-parts. Point-set topology begins the foundational aspects of topology and investigates notions inherent to topological spaces (fundamental illustrations involve compactness and connectedness). Algebraic topology attempts to measure degrees of connectivity by using algebraic constructs like homology and homotopy groups. The Geometric topology principally studies manifolds and their embeddings in other manifolds. An active region in geometric topology is low dimensional topology that studies manifold of four or fewer dimensions. This involves knot theory that is the study of mathematical knots

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