Hello, Athikan is here. Do you guys know that the donut and the teacup are similar? They have the same topological property. You guys have definitely heard about topology, the new trending term appearing in many articles. The shape of the universe, quantum computers, and advanced mathematics research papers contain this wonderful concept. So, what is topology? I am bringing a brief answer to this question.

Topology is the more abstract part of mathematics, which is both a very hard and a very easy subject in my graduate syllabus, with the reference book Topology by James R. Munkres. Literally, I struggled. And now I am learning topology again, and things are clearer. Also, I found a good introduction in one of the Soviet books, Introduction to Topology by Yu. Borsovich, N. Bliznyakov, Ya. Izrailevich, and T. Fomenko, printed by Mir Publications, Moscow, which I took from my professor’s desk and have not returned till now. I liked the way that book explains topology.
The base idea of topology arises from the concept that created graph theory two centuries ago: the problem of the Seven Bridges of Königsberg, which was solved by Euler. Wikipedia claims that many experts consider this problem as the birthplace of topology. And there are multiple people—Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann, and Enrico Betti—who contributed to the development of the theory. Research on topology began in the mid-19th century with investigations in function theory by Riemann. He developed new methods using geometric representations. After Riemann, Betti developed the notions of topology.
Topology was born from the great effort of the French mathematician Henri Poincaré at the end of the 19th century. Poincaré found topological concepts necessary for mathematics and made topology a profound theory, using it extensively in his research. His ideas and the problems he suggested are the pillars of the development of topology.

Even he mentions:

Quant à moi, toutes les voies diverses où je m’étais engagé successivement me conduisaient à l’Analysis Situs.

English translation:

As for me, all the various journeys on which, one by one, I found myself engaged were leading me to Analysis Situs.

I forgot to mention that topology was called Analysis Situs. A pretty cool name. And he defined Analysis Situs as:

L’Analysis Situs est la science qui nous fait connaître les propriétés qualitatives des figures géométriques non seulement dans l’espace ordinaire, mais dans l’espace à plus de trois dimensions.
L’Analysis Situs à trois dimensions est pour nous une connaissance presque intuitive. L’Analysis Situs à plus de trois dimensions présente au contraire des difficultés énormes; il faut, pour tenter de les surmonter, être bien persuadé de l’extrême importance de cette science.
Si cette importance n’est pas comprise de tout le monde, c’est que tout le monde n’y a pas suffisamment réfléchi.

English translation:

Analysis situs is a science which lets us learn the qualitative properties of geometric figures not only in ordinary space, but also in spaces of more than three dimensions. Analysis situs in three dimensions is almost intuitive knowledge for us. Analysis situs in more than three dimensions, on the contrary, presents enormous difficulties, and to attempt to surmount them, one should be persuaded of the extreme importance of this science. If this importance is not understood by everyone, it is because everyone has not sufficiently reflected upon it.

What is the qualitative property of geometric figures?

Consider a sphere, which can be stretched and compressed without being torn or having any two distinct points glued together. Such transformations of a sphere are called homeomorphisms. If two different results obtained by homeomorphisms are related in this way, they are said to be homeomorphic to each other. This means the qualitative property of the sphere is preserved under the homeomorphism. Or, the topological properties of the sphere are preserved under homeomorphism.
In the above example, the qualitative property or topological property of a sphere is connectedness. There are no separate regions in a sphere that we cannot connect by a continuous function. And we can easily say that a sphere and a hollow ball are not homeomorphic. They possess different topological properties. You can contract a ball into one of its points by changing it smoothly, but you cannot contract a sphere like that. And you can find that a volleyball and a cycle tire also have different topological properties. It feels like topology generalizes the physical properties of things around us.
And the famous teacup and donut question also involves topological properties. Having a hole is one of the topological properties. I can stretch some part of a donut and make a teacup. And topologically, a straw containing one hole is also homeomorphic to a teacup, a donut, and the tube of a cycle tire. Really, it is a very good generalization.

The research carried out by H. Poincaré is the starting point for one of the branches of topology: combinatorial or algebraic topology. This discusses the general properties of objects using algebraic structures such as groups, rings, and fields. The algebraic objects constructed by Poincaré are homology groups and fundamental groups. This development of topology led to the ideas of set-theoretic topology, which was developed by Georg Cantor and Hausdorff. Even Poincaré tried to generalize objects in spaces of more than three dimensions and discuss their topological properties.
This attempt led to the development of the concept of a topological space, the fundamental idea that pervades all of mathematics. The first fair definition of a topological space was given by Fréchet, Riesz, and Hausdorff. I tried to find their original definition on the web, but I did not get it.
The complete definition of a topological space was given by K. Kuratowski and P. S. Alexandrov. For this also, I did not get their original statements. If someone finds the original writings, don’t forget to share them in the comments.
Why is topology more significant? Because the concept arises from basic set theory. And it can explain real analysis, complex analysis, and some parts of algebra—more importantly, geometry. It can even discuss the shape of space itself in higher dimensions. It has a logic that can back major parts of mathematics, and it is a very abstract way of thinking about objects. That is why topology is more significant. It can open new pathways in science and innovation.