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Topology, often referred to as "rubber-sheet geometry," is a branch of mathematics that studies the properties of space that are preserved under continuous transformations, such as stretching and bending. This means two objects are considered equivalent in topology if they can be deformed into each other without cutting or gluing. For example, a donut and a coffee cup are topologically the same because they both have one hole.
The subject can be understood informally as how elements of a set relate spatially to each other. It's worth noting that the same set can have different topologies. Topology also includes the concept of "open sets" which are a collection of subsets within a set of mathematical objects.
In addition, topology has practical applications in many areas, including "network topology" in computer science, where it refers to the physical and logical arrangement of nodes and connections in a network.
Recent studies have explored the application of topology in understanding natural phenomena like weather patterns and in the field of high harmonic generation.
Below are some of the key topics typically studied in the field of topology:
These topics form the foundation of the study of topology, each with its unique principles, theorems, and applications.
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