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sammycaps
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Does anyone have any good reference to exercises concerning these topics? I would like to understand them better. Thank you.
mathwonk said:the point is to understand complicated spaces in terms of simpler ones.
i.e. an interval is simpler than a circle but a circle i q quotient o an interval.
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'a torus is moire complicated than a rectangle but a torus is a quotient of a rectangle,...
almost any space is a successive union of quotients of rectangles of various dimensions.
A quotient topology is a mathematical concept in topology that is used to create new topological spaces by "gluing" together existing spaces. It is created by taking a given space and identifying certain points or subsets as equivalent, and then defining a new topology on the resulting set of equivalence classes.
An adjunction space is a specific type of quotient topology, where the equivalence relation is defined by "gluing" a subspace to another space along a common boundary. This creates a new space that combines the properties of both the original spaces.
Quotient topologies and adjunction spaces are useful in mathematics because they allow for the creation of new spaces with desired properties by combining existing spaces. They also help to simplify the study of complex spaces by breaking them down into smaller, more manageable components.
Quotient topologies and adjunction spaces are used in various fields of mathematics, including algebraic topology, differential geometry, and functional analysis. They also have applications in physics, engineering, and computer science, particularly in the study of dynamical systems and data analysis.
One limitation of quotient topologies and adjunction spaces is that not all spaces can be constructed using these methods. Additionally, the resulting spaces may not always have the desired properties or may be difficult to study due to their complexity. Care must also be taken when defining the equivalence relation to avoid creating spaces with trivial or uninteresting topologies.