An Introduction to Topology

Topology Cover

 

Topology and Topological Space

[box title=”Topics” style=”default” box_color=”#005ce6″ title_color=”#FFFFFF” radius=”3″] [/box]

An Introduction to Topology

 

The word topology is derived from the Greek language. τόπος, ‘place, location’, and λόγος, ‘study’. Hence it is a study of objects, precisely geometric objects which remain constant (technically we call them invariant) if the object undergoes continuous deformation which is stretching it, twisting, bending it. So we study the shapes and sizes of different objects, however, the study of topology is much wider and broader than that of Euclidean geometry.

 

As we will see that the generalization of Euclidean space, the notions of distances and points are much wider. We always move towards generalization, from Newtonian mechanics to Einstein, from non-Euclidean geometry to Euclidean geometry, from plane surface to curved surfaces, and then to Riemannian manifolds… All these help us to encapsulate formulas and concepts.

 

The goal towards generalization is that we understand things in a much broader way. If one theory, one formula could give us solutions to many other things, why should we not go for that? You see in our daily world, how we generalize things. In one mobile phone if we get all the features then… A camera, a recorder, a talking device, a messaging device, everything is in one place. Mathematical models also frame the same thing. To make life easier, to encapsulate everything in one place. Topology deals with the same thing. The notion of geometry is much broader here and as we will see it contains various other features which makes the study of shapes much more effective.

 

Historically, it was Leonhard Euler who first, in his 1736 paper on the Seven Bridges of Königsberg, laid the foundation stone of topology. As a result, the polyhedron formula $V − E + F = 2$ (where $V$, $E$, and $F$ respectively indicate the number of vertices, edges, and faces of the polyhedron) came into play.

Mathematicians consider this to be the first formulation for the subject of topology.

Let us go ahead and first understand this formula.

 

$V$=vertices
$E$=edges
$F$=faces

 

Figure 1 - A tetrahedron
Figure 1 – A tetrahedron

 

In Figure 1 we have,

 

$V$=4
$E$=6 and
$F$ =4

 

By using Euler’s formula $V-E+F=2$ we get $4-6+4 =2.$ That means a tetrahedron satisfies this equation. So we call it the Euler characteristic of that shape. If you take into account, three dimensions, for any simple polyhedron or spherical polyhedra (why we call it simple polyhedron requires further discussion i.e. which can be ballooned to a sphere, for the time being, let us understand, the one whose faces are simple polygons) the Euler characteristic is $2$. You can experiment with this formula by using tetrahedrons or octahedrons. Not all surfaces will have the Euler characteristic of $2$. For example, simple polygons, that carry no hole, their Euler characteristic is $1$.

So, Euler gave us something through which we can identify the characteristics of different geometrical aspects. That means if the Euler characteristic of one object is $2$ then it should have some property if it is zero then something if it is $2$ then something and so on.

This is what topology does. It finds out the property of one geometrical object and hence tries to find the application of it to others if it satisfies the conditions.

This is a very basic and rough idea of what topology is. There are much more intricacies that we would slowly cover.

Next | Applications of Topology

 

See Also | More Topology Articles


Further References:

  • Topology – James R. Munkres
  • Topology without tears – Sidney A.Morris

Article Advisor:

  • Richard Sot (PhD. Mathematics, University of Rochester, Rochester, NY)

Website references:

  1. https://mathworld.wolfram.com
  2. https://www.wikipedia.org/
  3. https://math.stackexchange.com/

 

3 thoughts on “An Introduction to Topology”

Leave a Reply

Your email address will not be published.