Math Assignment Help >> Topology

Topology is an area of mathematics which is concerned with properties those are preserved under continuous deformations of objects, such as deformations which involve stretching, but no gluing or tearing. It emerged through the development of concepts from set theory and geometry, such as space, transformation and dimension.

The topology word is commonly used both for the mathematical discipline and for a family of sets with certain properties those are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms, which may be defined as continuous functions with a continuous inverse. For instance, the function y = x³ - homeomorphism of the real line.

Topology involves many other subfields. The most basic and traditional division or sub field within topology is point-set topology, that establishes the foundational aspects of topology and investigates concepts inherent to topological spaces for examples include compactness and connectedness; algebraic topology, that commonly tries to measure degrees of connectivity using algebraic constructs like as homotopy groups and homology; and geometric topology, that primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, like as graph theory and low dimensional topology, do not fit neatly in this division. Knot theory studies mathematical knots.

Topology, as a area of mathematics, may be generally defined as "the study of qualitative properties of certain objects known topological spaces those are invariant under certain kind of transformations called continuous maps, especially properties that are invariant under a certain kind of equivalence known homeomorphism."

The topology term is also commonly used to refer to a structure imposed upon a set of X, a structure which essentially characterizes the set X as a topological space by taking proper care of properties like as convergence, continuity and connectedness upon transformation.

Mathematical Definition

Let X is any set and let T is a family of subsets of X. Then T is known a topology on X if:

1. Both empty set and X are elements of T.

2. Any union of arbitrarily others elements of T is an element of T.

3. Any intersection of finitely others elements of T is an element of T.

If T is called a topology on X, then the pair (X, T) is known a topological space, and the notation X_T may be used to denote a set X endowed in the particular topology T.

The open sets in X are defined as the members of T; note that in general not all subsets of X need to be in T. A subset of X is called to be closed if its complement is in T i.e., its complement is open. A subset of X can be open, closed, both, or neither.

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