Topology studies properties of spaces that are invariant under any continuous deformation. It is sometimes called "rubber-sheet geometry" because the objects can be stretched and contracted like rubber, but cannot be broken.
For example, a square can be deformed into a circle without breaking it, but a figure 8 cannot. Hence a square is topologically equivalent to a circle, but different from a figure 8. Here are some examples of typical questions in topology: How many holes are there in an object?
How can you define the holes in a torus or sphere? What is the boundary of an object? Is a space connected? Does every continuous function from the space to itself have a fixed point?
Topology is a relatively new branch of mathematics; most of the research in topology has been done since The following are some of the subfields of topology. Algebraic topology sometimes uses the combinatorial structure of a space to calculate the various groups associated to that space. Topology is used in many branches of mathematics, such as differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis.
It is also used in string theory in physics, and for describing the space-time structure of universe. The central hub also acts as a repeater to make sure there is zero or minimal data loss during the transmissions. A star topology can be configured using a twisted pair, coaxial cable, or an optical fiber. Mesh topology is a widely used network model that has a point-to-point connection between each node in the network.
Every node or a device in a mesh network connects to other nodes directly and in a non-hierarchical manner. Every node in a mesh network can have a routing logic and transmission of data or information happens through that routing logic. This routing logic can be used to find the shortest distance to send some information from sender to receiver or the logic can be used to avoid using broken lines for data transmission. In the case of flooding, the same data is transmitted to each node in the network.
Therefore, no routing logic is required in case of flooding mesh networks. Loss of data is highly unlikely as every node will have the same data with them. This makes it robust and fault-tolerant. However, this also increases the load on the network. When a network topology is formed by integrating two or more topologies together, this results in a hybrid topology.
When configured properly, hybrid topologies can provide the best of all the network topologies. Hybrid topologies are easy to scale and expand. However, they might need higher costs and more operational efforts to configure and maintain. A tree network topology is one of the most common examples of a hybrid topology. It is also referred as a star-bus network topology in which star networks are interconnected with one another using a bus network.
In a tree topology, nodes are connected with one another in a hierarchical manner and are therefore also known as hierarchal topology. Choosing the right network topology depends on multiple factors such as the number of nodes to be involved in the network, geographical distance between the nodes, finances, maintenance, operational flexibility, and more.
Every topology we discussed above has its own advantages and disadvantages. Therefore, the key to build and configure the right networking model is subjective.
For any company, it is very important to first gather all the requirements and needs before adopting any particular network topology. Sukesh is a Technology Consultant and Project Manager by profession and an IT enterprise and tech enthusiast by passion. He holds a Master's degree in Software Engineering and has filled in various roles such as Developer, Analyst, and Consultant in his professional career.
He holds expertise in mobile and wearable technologies and is a Certified Scrum Master. Was this article written in ? No one in their right mind would use a bus or ring topology today. What are interesting not too hard applications of topology in other areas of mathematics? Well, an idea would be to talk about results of Real Analysis and how topology generalises them, or how topological methods make their proofs much easier.
Tychonoff's theorem might also be a nice one, or the converse of the Closed Graph Theorem. My guess is that you should connect it to metric spaces mostly, for they will be mostly familiar with that.
I can keep going but I am afraid that it would be pointless if any of my previous suggestions is not readily utilisable. Best of luck!
My first exposure to topology, before I even realized that it was topology, was in Zorich's Mathematical Analysis I. It was through the definition of a limit over a base. After defining limits at a real number and at infinity in the standard way, which I was already familiar with, the author suddenly introduced a concept I'd never heard of: a base.
I don't have the book to hand, but I'm pretty sure that was all the properties. At first I was confused, but as he defined the notion of a limit over an arbitrary base, I realized this was the solution to something that had always bugged me about limits. And here it was: the single, unified definition I'd always wanted. I became even more impressed when I realized the same notion even covered limits of sequences.
He explained that the definition came by simply observing that properties 1 through 3 were really the only ones used in proving the key properties of limits.
Flipping back, I realized he was correct, and I understood: mathematical definitions are sometimes obtained by distilling a theory. You systematically gather up everything you used to prove the theorems, and obtain a precise description of all of the objects that satisfy those theorems or at least, some sufficient conditions.
Most importantly, my understanding came from seeing a real application of topology in mathematics, not just some vague "intuition" of what the definition "means". In functional analysis, the notion of weak topology is a great example of application of "weird" not metrizable topology : arguments of weak compacity provide very quickly the existence in a lot of variational problems including for example the Lagrangian formulation in classical mechanics or limited relativity.
The "Hairy ball" theorem has a funny consequence : at every given time there are two antipodal points at the surface of the Earth that have exactly the same temperature and pression. Thus the intermediate value theorem provides an existence theorem in cases where the theorem of Abel tells us there is no formula in radicals for a root of a polynomial. Adding the mean value theorem we get estimates on the number of solutions. Similar two dimensional versions allow the fundamental theorem of algebra to be proved, again guaranteeing solutions to polynomial equations and counting their number.
The general theory of degree of mapping generalizes this technique and even extends to infinite dimensional spaces. The theory of singularities of vector fields is another example of applying topology to questions of existence of solutions. Cauchy's theorem also allows the concept of simple connectivity to be invoked to conclude existence of complex logarithms in certain regions.
The list is long
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