What is Topology?

Topology Cover

Topology and Topological Space

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What is Topology?

 

Topology is the study of shapes, which can be stretched, squished, and otherwise deformed keeping ‘near points’ together.

Now ‘near points’ or ‘nearby’ is an undefined term that roughly means a small neighbourhood.

 

Figure 2 - A set V in the plane of a point p
Figure 2 – A set V in the plane of a point p

In Figure 2 we can see a set which is labelled as $V$ in the plane is a neighborhood of a point, labeled $p$ if a small disc is contained in $V$.

Suppose $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V$ of $X$ that includes an open set containing $p$. Mathematically, it is this:

$p∈U∈V$

Don’t worry about these terms as it will become clear as we proceed. You can also think of a point that moves around a bounded area, without leaving that set.

 

Figure 3- Homeomorphism and inverse function
Figure 3- Homeomorphism and inverse function

 

The above figure shows under homeomorphism the property is preserved. Right now, let us think of homeomorphism as a correspondence between two figures such the function and is preserved. As you can see in Figure 3, the cup morphs to a torus and then in Figure 3, the reverse takes place. Tearing and gluing are not allowed. However, that is not true. There is something called Dehn Twist – a topological move that cannot be achieved without cutting and pasting.

 

Figure 4 - Congruency of angles and sides
Figure 4 – Congruency of angles and sides

 

Classically speaking  Figure 4 shows what is called congruency in Euclidean geometry. Two shapes are congruent when one can be mapped to another. You might be wondering why are going back to Euclidean geometry?

 

Figure 5 - Congruency retained under rotation
Figure 5 – Congruency retained under rotation

 

Figure 5 shows that, even under rotation, the angles do not change. However, in Euclidean geometry, no deformation is allowed. Figure 5 shows that when rotated, they preserve their congruency.

 

Figure 6 - Polygonal shapes to any shapes
Figure 6 – Polygonal shapes to any shapes

 

Figure 6 shows that from the regular polygonal shapes, we are moving towards any shapes and sizes. This is how the evolution from Euclidean geometry moves towards topology, as now we are trying to formulate something, through which we can study any shape and size.

 

Figure 7 - Homeomorphism between a cup and a donut
Figure 7 – Homeomorphism between a cup and a donut

 

So, the regular polygons are replaced by any shape, yet maintaining their structure, what is known as. It means that some amount and some part of the function is preserved and does not change as you can see in Figure 7, the torus or the donut, represents a coffee cup.

if a space $X$ possesses a property, every space, homeomorphic to $X$ has that property.

The above definition is a little more technical. So, it is evident that the property of one object is isomorphic $≅$ to another.

 

Figure 8 - Isomorphism
Figure 8 – Isomorphism

 

Figure 7 is quite self-explanatory. Some part of the property of the circle is $≅ $ to the square and so on.

 

 

Figure 9 - Preservation of properties
Figure 9 – Preservation of properties

 

Figure 9 clears the concept better, how the property of each of the geometrical shapes is preserved. So, although geometrically they are different, yet topologically they are the same.

 

 

Figure 10 - Bi-directional mapping
Figure 10 – Bi-directional mapping

 

Figure 10 shows how the torus turns into another torus of different shapes. So, you see, the topological invariance is maintained which can be written as:

Bi-directional mapping of one shape to another which preserves the notion of points being near to each other.”

Recall, at the beginning we talked about the notion of nearness which is now reflected here.

So, if you have a rubber band or clay, you can form a circle, ellipse, or a square. These are very different geometrically but topologically they are the same.

  1. The notion of closeness is now:
  • Continuity
  • Convergence
  • Compactness and
  • Connectedness
  1. The notion of invariance is preserved under continuous deformation
  2. Formalizes the notions of ‘nearness’ and ‘continuity’
  3. Generalizes the concept of analysis/calculus

 


 

Topology: Technical Definition

 

Let $X$ be a non-empty set. A set $T$ of subsets $X$ is said to be a topology if:

(i) $X$ and the empty set, $\phi$ belongs to $T$
(ii) the union of any finite or infinte number of sets $T$ belongs to $T$ and
(iii) the intersection of any two sets $T$ belongs to $T$

The pair $(X,T)$ is called a $topological space$

Three things emerge from the above definition:

  1. $\cap $ – $intersection$
  2. $\cup $ – $union$ and
  3. Open sets

Before we understand the technical definition, let us step back around the middle of 4th. century BC where a person, framed some mathematical formulations, which laid the foundation for fundamental geometry.

 

Next | What is Euclidean Space?

 

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/
  4. https://www.researchgate.net/publication/343635292

 

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