Theoritical Talk About Topology

Received: 25-Jan-2021 Accepted Date: Jan 27, 2021; Published: 29-Jan-2021, DOI: 10.37532/2752-8081.21.5.19

Citation: Michael Th. Rassias. Theoritical talk about topology. J Pure Applied Mathematics. 2021; 5(1):1-1.

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Mathematicians associate the emergence of topology as a distinct field of mathematics with the 1895 publication of Analysis Situs by the Frenchman Henri Poincare, although many topological ideas had found their way into mathematics during the previous century and a half.

The definition of Topologyis it’s a relatively new branch of mathematics as most of the research in Topology has been processing since 1900s and is used in fields of string theory for describing the space-time structure of universe.

We now consider a more general case of spaceswithout metrics, where we can still make sense of the notions of open and closed sets. These spaces are called topological spaces.

In topology a continuous function is often called a map. There are 2 different ideas we can use on the idea of continuous functions.

Homeomorphism:A homeomorphism is a function f: X→Y between two topological spaces X and Y that is a continuous bijection, and has a continuous inverse functionf−1.

Retraction:

A retraction is homotopicequivalence between a space X and a subspace

Y(Y⊆X).

When dealing with retractions we assume we are using the Euclid’s Topology.

Some of the subfields of topology are as follows:

Generalised topology normally considers local properties of spaces, and is closely related to analysis.

It generalizes the theoreticalcontinuityto brief topological spaces, in which limits of sequences can be existing.

Combinatorial topology assumes the global properties of spaces, found from a network of vertices, edges, and faces.

This is the ancient branch of topology, and reverts back to Euler.

Algebraic topology swaps a topological problem into an algebraic problem that is faithfully easier to getsolved. Algebraic topology sometimes uses the combinatorial structure of a space to calculate the various groups associated to that space.

Differentiation topology is useful for learning properties of vector fields, such as a magnetic or electric fields, etc.