Topology and Topological Space[box title=”Topics” style=”default” box_color=”#005ce6″ title_color=”#FFFFFF” radius=”3″]
- Introduction to Topology
- Applications of Topology
- What is Topology?
- What is a Euclidean space?
- Euclidean Axioms to Topological Axioms
- What is an Open Set & Open Interval?
- Who was Pavel Sergeyevich Alexandrov?
- Proof of Topology and Topological Space through Axioms
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.
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:
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.
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.
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 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 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.
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 7 is quite self-explanatory. Some part of the property of the circle is $≅ $ to the square and so on.
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 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.
- The notion of closeness is now:
- Compactness and
- The notion of invariance is preserved under continuous deformation
- Formalizes the notions of ‘nearness’ and ‘continuity’
- 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:
- $\cap $ – $intersection$
- $\cup $ – $union$ and
- 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
- Topology – James R. Munkres
- Topology without tears – Sidney A.Morris
- Richard Sot (PhD. Mathematics, University of Rochester, Rochester, NY)
Shounak Bhattacharya is working as a Director, Training and Placement Department for Asian College of Teachers, Thailand. Apart from training, he is a researcher as well as a teacher in the area of
- Tensor analysis,
- General theory of relativity,
- Differential geometry and
- Introductory topology.
Shounak has been researching and educating the global crowd in the area of building career opportunities through teaching, researching, and communicating. Working with Asian College of Teachers, his main focus has been increasing employability factors among young students. The path leads to many factors including teaching, researching, communicating as well as creating a unique model which would employ the right person for the right job.
Relentlessly working in simplifying difficult mathematical concepts, Shounak’s interest revolves around Einstein’s general theory of relativity and creating structures, models, videos to educate the mass. He has been in touch with Prof. Lohiya, a former student of Dr. Stephen Hawking, and has conducted interviews regarding his experience working with Stephen Hawking.
He is working under the guidance of Dr.Richard Sot, Ph.D. Mathematics, University of Rochester, NY and Dr.Santosh Karn, Ph.D., Physics from Delhi University.
He has held numerous live interviews with researchers, educators, teachers, and dignitaries around the globe primarily,
- The United Kingdom,
- Australia and
He is a professional translator of the Arabic language, certified from RMIC. He is currently into Persian-Urdu translation of literary texts and creating videos on complex mathematical concepts.