Types of Topology

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Types of Topology

   

  
General or point-set Topology:

It generalizes the basic concepts such as connectedness, continuity, compactness by using definitions from set theory and lays the foundation for the other branches. Changing the topology can change which functions are continuous and which subsets are connected or compact. Topological spaces have metric spaces as an important class as they simplify many proofs.  

 

  general topology-coffee cup and doughnut

  

Algebraic Topology:

It associates the algebraic objects with topology. In this algebraic structures are used as tools to study the variations of geometric objects. The basic idea is to consider two spaces to be identical if they have the same shape. Sometimes, the topologically isomorphic shapes seem different geometrically, like a cuboid and sphere but their algebraic properties are similar, so it becomes easier to study it by converting these problems to algebraic ones. 

 It gives the convenient Proof of Nielsen-Schreier theorem, any subgroup of a free group is again a free group, The fundamental algebraic objects, and structures used in this are groups and vector spaces. 

 

algebraic topology- sphere

   

Differential Topology:

It deals with the smoothness associated with every point in space. It considers differential functions on smooth surfaces;  it is also quite similar to differential geometry and together they can form the geometric theory of differential manifolds. Useful tools are provided in this to study and extract the underlying properties of complex spaces. The vector bundle is the link between algebraic and differential topology.  

differential topology-vector bundle

  
Combinatorial(Geometric) Topology:

It is the subset of algebraic topology but uses methods of combinatorics, hence, relies totally on counting. It deals with the local properties of spaces and studies the geometry of objects in higher dimensions.


The combinatorial structure of space is used in algebraic topology to calculate several groups associated with that space. It is so old that even Euler has worked on it and gave the famous Euler’s polyhedron formula which says that for any object to be topologically equivalent to sphere {F-E+V} should be equal to 2, which is called Euler’s characteristics, where F, E, V are the no of faces, edges and vertices respectively.


Euler's formula of polyhedron

 

 
Location: India
Hey there, This is Hifza Siddiqui, an aspiring Mathematician pursuing masters from the University of Mumbai. I'm writing this blog to share the knowledge and the way I understand mathematics. I hope it helps you to get a better insight. My blog's frequency will be twice per week.

Comments

  1. UnknownJuly 17, 2020 at 8:10 PM

    Typical Topological Topology✌️. Well understood, good explanation.
    But next time explained with few examples please ☺️

    ReplyDelete

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