What is Topology?
Today our topic is what is topology?
We often listen to the word “Topology”. We too read it at the level of BSc or MSc or it is also studied when change in shapes are discussed in any field of life whether that is a mathematical problem or artistic problem but have you ever find the real meaning if the topology? That actually what is it?
The word topology means “the study of geometrical properties and spatial relations unaffected by the continuous change of shape or the size of the figure. Or we can define it as “The way in which constituent parts are interrelated or arranged.
Topology and Examples:
If I expand the meaning of topology by some examples like where geometry is concerned with distances (lines, angles, curvature, etc.) topology is concerned with the abstract notions of continuity, connectedness, and compactness.
Claude and Ryan:
If I take the meaning from Claude and Ryan about topology then they said that “Topology can be used to abstract the shape and behavior of proteins, enzymes, and DNA”. And this definition shows that the topology is basically the talk of shape and sizes as biology is concerned with the human’s body and human body consists of all organs which have shapes. If I take another example to make you more clearly about the topology then particularly quantum mechanics and strings theory where tiny variations in the shape matter are studied and that is topology.
Nowadays TDA (Topological Data Analysis) is far the most prominent application of the topology in real life but now the question arises what is TDA? And I am here to solve your problem that if any data is mapped to some kind of representation in Euclidean Space, then that data has a shape and study of that data is TDA. If something has shape than topological techniques can be used to study the intrinsic properties of that shape. The basic of TDA is to study the properties of data which are invariant to dimensionality noise and other factors which may cloud our interpretation of the data.
Scans and Topology:
If I quote some more examples to clear you about the word topology? Or when how it is used or studied? Then let me give you more examples like suppose we do some sort of scan and that can be like CT scan or MRI or ultrasound or any other type of an appropriate scan for example of our stomach area, we may accept such as intestines, pancreas, liver, etc. which can be thought of as “components” of this data which we are interested in. In reality, we only get data which has little use in its raw or unprocessed form which need to be processed so as to extract such useful information. To extract such information, methods of computational topology are used which can then be used to “separate “or “isolate” these components of this data to get useful information about these organs i.e., components of the data simple.
“Topology can be thought of as abstracting geometry by removing the concept of the distance.”
Cosmologist uses a lot of topologies when they study the structure of the universe, now the questions arise that why they need to study the topology? Basically, it is a study of data with shapes and the entire universe has some kind of specific shape.
So, we are here with some sort of example like:
“Topology is the branch of mathematics which is concerned with the deformation of an object and space but geometric objects that don’t change when bent or stretched.”
Combined with algebraic geometry, abstract algebra, differential equations, and probability, topology is now used in many branches of physics to understand the exotic state of matter (state of matter that is not solid, not liquid, mot gas that has very unusual physical characteristics.
Let us give a famous example as I am always here to teach you the topic with an example. As we have listened “A topologist is a man who does not know the difference between a doughnut and a coffee cup”.
This is because topology is a study of geometrical objects without considering things like length, angles, and shapes. A coffee cup can be transformed into a donut by squashing, stretching and twisting, so the two objects are topologically equivalent. However, a ball cannot be turned into a donut without turning or cutting so they are topologically different.
For mathematicians the definition of topology is:
“Let X is a non-empty set. A family T of all the subsets of X is called a topology on X if it follows the following three properties like:
1- Any finite intersection of the members of T is again a member of T (and all the members in T is from the open set)
2- Arbitrary union of members of T belongs to T.
3- The empty set and the given set are the members of T.
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The elements of T are called points in topological space (if X is non-empty set and T be a topology on X. The pair (X, T) is called topological space) and the members of the T is called open set.
Examples of Space and Real life Use of Topology:
Further, we have many types and examples for the topological space but here I was only discussing the definition of the topology and few real-life examples (no example is given from data because that was not the topic of concerned).
In the end, as it is the part of life and we use it without thinking but actually that is used in life. I want to wind up by saying that:
“Topology is concerned with the geometrical objects without studying there shapes and deformation of objects without considering the methods and it is the study of the space.”
Meaning according to my search: