Question on the properties of a manifold

In summary, a manifold must fulfill three requirements according to the given text: it must be Hausdorff, locally euclidian, and have a countable basis of open sets. While most books explicitly show the locally euclidian character of a manifold, the embedding in En is used to assume the hausdorff and countable basis requirements. While it is clear that Rn meets the hausdorff condition, it is not immediately apparent that it also has a countable basis of open sets. However, it can be proven by starting with R and using the set of rational numbers as center points and rational radius intervals as open balls. It is important to note that these technical properties may seem puzzling, but they are
  • #1
majutsu
12
0
According to my text, a manifold should be 1) Hausdorff (that is t-2 separable, so there are disjoint open sets which are neighborhoods for any two points x and y), 2) locally euclidian (that there is a neighborhood U of a point x that is homeomorphic to an open subset U' of Rn (the RxR...xR cartesian product) and 3) has a countable basis of open sets.

In most books, when the set out to illustrate something is a manifold, they usually explicitly show the locally euclidian character. Then the embedding of that manifold in En (euclidian n-space) is used to assume the hausdorff and countable basis (paracompact) requirements. The hausdorff assumption, I have no trouble with, as Rn clearly meets that condition as I understand it above. But I don't assume that En or Rn have a countable basis of open sets.

Can someone prove to me that Rn has a countable basis of open sets?
 
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  • #2
try to prove it yourself. start with R. take the most natural countable set of center points i.e. the rationals., and take the most natural countable collection of open balls, i.e. rational radius intervals.

a piece of advice however:

these technical properties have no importance whatever, except that they are true; i.e. if you cannot prove them, but are willing to assume them, you are still going to get everything possible from the subject that is of importance.

of course i understand a curious poerson has difficulty by passing a puzzling statement.
 
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  • #3


Yes, it can be proven that Rn has a countable basis of open sets. This is one of the properties of a topological space that is often assumed and not explicitly stated.

To prove this, we first define a basis for a topological space. A basis for a topological space X is a collection B of open sets such that every open set in X can be written as a union of elements of B. In other words, the elements of B "generate" all the open sets in X.

Now, let's consider the standard topology on Rn, which is the topology induced by the standard metric on Rn. In this topology, the open sets are defined as follows: a set U is open if and only if for every x in U, there exists a positive real number r such that the open ball of radius r centered at x is completely contained in U.

Using this definition, we can construct a basis for Rn as follows: for every positive integer n, we can consider the set of all open balls in Rn with rational radii and rational center coordinates. This set, let's call it Bn, is countable since the rationals are countable.

Now, we claim that B = {Bn | n is a positive integer} is a basis for Rn. To see this, let U be any open set in Rn. Then for every x in U, there exists a positive real number r such that the open ball of radius r centered at x is completely contained in U. Since the rationals are dense in the reals, we can always find a rational number r' that is arbitrarily close to r. Therefore, the open ball of radius r' centered at x is also contained in U. This means that for every x in U, we can find an element of B that contains x and is completely contained in U. Hence, U is a union of elements of B and B is indeed a basis for Rn.

Therefore, Rn has a countable basis of open sets, as claimed.
 

Related to Question on the properties of a manifold

What is a manifold?

A manifold is a mathematical concept that describes a space that looks like Euclidean space on a small scale, but may have a more complicated structure on a larger scale. In simpler terms, it is a topological space that appears flat and smooth when examined locally, but may have a curved or distorted shape when viewed globally.

What are the properties of a manifold?

The properties of a manifold include being locally Euclidean, meaning it looks like Euclidean space on a small scale, and being connected, meaning there is a path between any two points on the manifold. It also has a dimension, which is the number of coordinates needed to specify a point on the manifold. Other properties may include smoothness, orientability, and compactness.

How are manifolds used in science?

Manifolds are used in various fields of science, such as physics, mathematics, and computer science. In physics, they are used to describe the space-time continuum and the curvature of the universe. In mathematics, they are used to study geometric structures and to solve problems in topology. In computer science, they are used for data visualization and pattern recognition.

What is the difference between a smooth and a non-smooth manifold?

A smooth manifold is one that can be described by a set of smooth functions, meaning they are infinitely differentiable. This allows for a smooth transition between points on the manifold. On the other hand, a non-smooth manifold may have abrupt changes or discontinuities, making it more difficult to study and work with mathematically.

Can manifolds exist in more than three dimensions?

Yes, manifolds can exist in any number of dimensions. In fact, some mathematical theories, such as string theory, require the existence of manifolds in higher dimensions for their equations to work. However, it can be difficult to visualize and understand manifolds in higher dimensions, as our brains are limited to perceiving three-dimensional space.

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