What is topology?
Topology is a particular area of mathematics that focuses on studying geometrical properties, which are unaffected when subjected to continuous deformations, such as shrinking, expanding, stretching, twisting, and bending. These deformations, however, do not include actions such as tearing, gluing, opening holes, closing holes, or passing through itself.
In a topological sense, two objects are considered to be equivalent if, by these deformations alone, they could be deformed into one another. By these standards, a circle is considered to be topologically equivalent to an ellipse (a 2D oval), as you can make an ellipse by simply stretching a circle. Similarly, a sphere is topologically equivalent to an ellipsoid (effectively a 3D oval), since you can manipulate a sphere, via topological deformations, into an ellipsoid.
Whilst similar to geometry in many ways, topology differs in the fact that numerical quantities, such as widths, angles, and heights, are often shared between geometrically equivalent objects. In contrast, however, objects that are topologically equivalent often share more qualitative similarities.
Topology is actually a very extensive area of study, whereupon there are many different branches of topology, such as:
General topology
Algebraic topology (which uses algebra to study topological spaces)
Differential topology (which uses differentiation on differentiable manifolds)
Geometric topology
The underlying ideas and motions for topology can be traced back to Gottfried Leibniz, who was a German mathematician in the 17th century. However, the branch's first theorems are more heavily linked to Leonhard Euler and his Seven Bridges of Königsberg problem. This was a mathematical problem that cemented the foundations of graph theory and heralded the idea of topology.
Figure 1: a topological shape (the Klein bottle) that is equivalent to the connected sum of two crosscaps
General topology and its basic concepts:
General topology is the foundation for most of the other topological branches – in itself, general topology deals with basic and fundamental set-theoretic constructions and definitions applied to topology.
One of the most basic areas of topology to study is topological spaces. Topological space is simply geometrical space in which closeness is defined, but numerical distance cannot be measured. Concepts that are rudimentary to topology as a whole, such as continuity, compactness, and connectedness, can be explained with general topological ideas.
Continuity: can trace one point to another to another to another
Compactness: sets that can be covered by a finite amount of random small sets in size
Connectedness: where disjoint, non-empty open subsets cannot be used to represent a topological space
Applications of topology:
Surprisingly, the study of topology can be used in many different disciplines, including biology, computer science, and physics. From a biological perspective, topology can be used to study aspects of living systems, especially molecular structures, such as folded proteins and nucleic acids, and nanostructures. Topology is also prevalent in various areas of physics, such as mechanical engineering, condensed matter physics, and, more specifically, quantum field theory.
References:
Wikipedia contributors. (2021b, April 10). Topology. Wikipedia. https://en.wikipedia.org/wiki/Topology#General_topology
Topology - Homeomorphism. (n.d.). Encyclopedia Britannica. https://www.britannica.com/science/topology/Homeomorphism
Wikipedia contributors. (2021c, April 28). Connected space. Wikipedia. https://en.wikipedia.org/wiki/Connected_space
Figure 1 reference:
Wikipedia contributors. (2021a, March 5). Klein bottle. Wikipedia. https://en.wikipedia.org/wiki/Klein_bottle
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