Demonstrating Openness in R^n Sets

[proplaya201]

Homework Statement

how do you show a set is open in R^n?

Homework Equations

The Attempt at a Solution

 

[CompuChip]

Depends on which course you are taking.
In analysis you could show that for every point in the set there is some small ball completely contained inside the set.
In topology you could work from the definition of open set, use a basis, show that the complement is closed, or even use some more elaborate theorem.

So please be a little more specific

 

[proplaya201]

im reading rudin's book: principles in mathematical analysis, ad we are talking about metric spaces, ie topology. so can you expand on you second approach to the problem please?

 

[HallsofIvy]

The same way you prove almost anything: use the definition of open set. What is the definition of open set you are using?

 

[proplaya201]

a set is open if every point in the set is an interior point. now i know that but i am having difficulty proving it.

(every point being an interior point that is)

 

[CompuChip]

So the proof would start like: "Let x be any point in the set ..." and then shows that x satisfies the definition of an interior point.

What is the definition of an interior point?

 

[proplaya201]

then there exists a neighborhood of x such that neighborhood of x is contained in the set