Show that a hyperplane in R^n is a closed set

[royzizzle]

Homework Statement

Show that a hyperplane in Rn is a closed set

Homework Equations

The Attempt at a Solution

I was thinking maybe try to prove that the complement of the hyperplane is open?

 

[LeonhardEuler]

I was thinking maybe try to prove that the complement of the hyperplane is open?

That's a good start. Now you need a description of a hyperplane to work with. How would you describe it?

 

[royzizzle]

That's a good start. Now you need a description of a hyperplane to work with. How would you describe it?

alright. given an x0 let x be a plane that passes through x0 and let x-x0 be orthogonal to a. and define the hyperplane to be the set of x such that a dot x = a dot x0

so the complement would be the set of x such that a dot x != a dot x0

so (a1 +...+ an)((x1-x01) +...+ (xn - x0n)) != 0

what should i do to prove that the set x is open?

 

[LeonhardEuler]

alright. given an x0 let x be a plane that passes through x0 and let x-x0 be orthogonal to a. and define the hyperplane to be the set of x such that a dot x = a dot x0

so the complement would be the set of x such that a dot x != a dot x0

so (a1 +...+ an)((x1-x01) +...+ (xn - x0n)) != 0

what should i do to prove that the set x is open?

Hmm, I don't follow that description. It might be easier to define it with a set of linear equations. Then the compliment is just the set of elements that don't solve each of those equations.

How to prove openness? What is the definition of an open set in R^n?