Topology- Hyperplane proof don't understand

In summary: Now I'm confused about that. Any point y in the ball around x of radius r is given by y=x+a where 'a' is any vector whose length is less than r. An awful lot of them don't satisfy a.y=0. Can you find just one?If a is a vector in R^3, then S is a plane. If I place an open ball on any point on the plane it would include points that are not on the plane. So S interior would be empty.
  • #1
zeion
466
1

Homework Statement



Let [tex] \mathbf {a} \in R^n [/tex] be a non zero vector, and define {[tex] S = \mathbf {x} \in R^n : \mathbf {a} \cdot \mathbf {x} = 0 [/tex] }. Prove that S interior [tex] = {\o} [/tex]


Homework Equations





The Attempt at a Solution



Intuitively I understand that if a is a vector in R^3, S would be a plane. And if I place an open ball on any point on the plane it would include points that are not on the plane. So S interior would be empty.

But I'm not sure how to prove this rigorously?
I'm supposed to somehow use the hint that a hyperplane in R^n will have 1 dimension less than R^n (like the plane in R^3 has 2 dimensions)
But I am confused about the dimension of the plane? All vectors on a plane in R^3 will have 1 (the same) component = 0? So does this mean that the dimension of the plane is 2?
 
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  • #2
Can I just show that there is a point in the ball that has n components ?
Instead of n-1
 
  • #3
zeion said:
Can I just show that there is a point in the ball that has n components ?
Instead of n-1

If a.x=0 you can pretty easily find a point y in any ball around x such that a.y is not zero.
 
  • #4
I'm a little confused about that.
Do I need to differentiate between a point and a vector?
Like if I show that there are points in the ball that are not on the plane, then do I need to specify that the vector corresponding to that point is not perp a ?
 
  • #5
zeion said:
I'm a little confused about that.
Do I need to differentiate between a point and a vector?
Like if I show that there are points in the ball that are not on the plane, then do I need to specify that the vector corresponding to that point is not perp a ?

Now I'm confused about that. Any point y in the ball around x of radius r is given by y=x+a where 'a' is any vector whose length is less than r. An awful lot of them don't satisfy a.y=0. Can you find just one?
 

Related to Topology- Hyperplane proof don't understand

1. What is topology?

Topology is a branch of mathematics that studies the properties of objects that do not change under continuous deformations, such as stretching and bending.

2. What is a hyperplane?

A hyperplane is a flat subspace of a higher-dimensional space. In two dimensions, it is a line; in three dimensions, it is a plane; and in higher dimensions, it is a subspace with one fewer dimension than the original space.

3. What is a proof in topology?

A proof in topology is a rigorous mathematical argument that demonstrates the truth or validity of a statement or theorem within the field of topology. This often involves using axioms and logic to logically deduce the desired result.

4. What is the hyperplane proof in topology?

The hyperplane proof in topology is a method of proving that two topological spaces are not homeomorphic (meaning they cannot be continuously deformed into each other) by showing that they have different numbers of hyperplanes. This proof relies on the topological invariant of connectedness.

5. Why is the hyperplane proof important?

The hyperplane proof is important because it allows mathematicians to distinguish between topological spaces and understand their topological properties. It also helps in identifying topological invariants, which are properties of spaces that do not change under continuous deformations and are useful in solving more complex problems in topology.

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