Introduction to Mainfolds Help

In summary, the conversation discusses the differences and relationships between Euclidean space, topological space, metric space, and vector space. It also suggests referring to Wikipedia for exact definitions. The conversation provides a brief explanation of metric and topological spaces, and suggests asking a specific question about vector spaces. Finally, it mentions the definition of n-dimensional Euclidean space.
  • #1
mundo44
5
0
I am taking a class on Manifolds and whilst reading the book I came about some terminology that has got me confused.

What is the differences/relations between these terms?

Euclidian Space
Topological Space
Metric Space
Vector Space

I thought that a vector space was based on the premise that each coordinate described by its values are of an Euclidian Space, but the book I am reading acts as though they are completely two different things. Also, the other terms are used in Differential Geometry and I have no experience with them. So could you guys clarify, I would appreciate it a lot! :)
 
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  • #2
mundo44 said:
I am taking a class on Manifolds and whilst reading the book I came about some terminology that has got me confused.

What is the differences/relations between these terms?

Euclidian Space
Topological Space
Metric Space
Vector Space

I thought that a vector space was based on the premise that each coordinate described by its values are of an Euclidian Space, but the book I am reading acts as though they are completely two different things. Also, the other terms are used in Differential Geometry and I have no experience with them. So could you guys clarify, I would appreciate it a lot! :)
I suggest Wikipedia for the exact definitions.

A metric space is a set equipped with a function that assigns a non-negative real number that you can think of as a "distance" to each pair of points in the set. This function is called a metric. For each x in X, and each r>0, the set [itex]B(x,r)=\{y\in X|d(x,y)<r\}[/itex] is called the open ball around x with radius r. A set that's equal to a union of open balls is said to be open. The most important thing the metric is used for is to define limits of sequences, but the definition can also be expressed in terms of open sets: Suppose that S is a sequence in X. A point x in X is said to be a limit of S, if every open set that contains x contains all but a finite number of terms of S.

The metric is also used to define what it means to say that a function is continuous, but it turns out that continuity can also be defined entirely in terms of open sets: [itex]f:X\rightarrow Y[/itex] is said to be continuous if [itex]f^{-1}(E)[/itex] is open for each open [itex]E\subset Y[/itex].

These two observations provide the motivation for topological spaces. A topological space is a set together with a specfication of which of its subsets are to be called "open". (The specification must satisfy a short list of conditions). You can use that specification (the "topology") to define limits and continuity, without using a metric.

Vector space...you probably know the definition already, so consider asking a more specific question.

Euclidean space...There are several inequivalent definitions. I think what you need is the simplest one: n-dimensional Euclidean space is just the set [itex]\mathbb R^n[/itex] with the standard vector space structure.
 

What is an Introduction to Manifolds course?

An Introduction to Manifolds course is a higher level mathematics course that focuses on the study of manifolds, which are geometric objects that can be described using the concepts of calculus and linear algebra. It is typically taken by students pursuing a degree in mathematics, physics, or engineering.

What topics are covered in an Introduction to Manifolds course?

Topics covered in an Introduction to Manifolds course include topological spaces, smooth manifolds, tangent spaces, vector fields, differential forms, integration on manifolds, and more. It also introduces students to the fundamental concepts and techniques used in manifold theory.

What are some real-world applications of manifolds?

Manifolds have many real-world applications, including in physics, engineering, computer science, and statistics. They are used to represent physical systems, such as the shape of the universe, and to analyze complex data sets, such as images and brain activity. They are also used in robotics, machine learning, and computer graphics.

What mathematical background is required for an Introduction to Manifolds course?

Students should have a strong foundation in advanced calculus, linear algebra, and multivariable calculus before taking an Introduction to Manifolds course. They should also be comfortable with abstract mathematical concepts and proofs.

What can I do with a knowledge of manifolds?

A knowledge of manifolds can open up many opportunities in various fields, including research in mathematics and physics, data analysis, and software development. It can also serve as a strong foundation for further study in areas such as differential geometry, topology, and mathematical physics.

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