- #1

Math_Frank

- 27

- 0

## Homework Statement

The Premise:

Here One must prove that that R^n and Ø are the two subsets of R^n, which is both open and closed. You must that these are the only subsets of R^n which has this property!

Let [tex]X \subseteq \mathbb{R}^n[/tex] be a subset, which is both open and close, and here we must prove that if either X = R^n or X = \emptyset. Thusly [tex]X \neq R^n[/tex] and [tex]X \neq \emptyset.[/tex]

Prove that this assumption leads to a contradiction:

Let [tex]Y = \mathbb{R}^n \setminus X[/tex] and show that Y is both open and closed and not empty!

## Homework Equations

## The Attempt at a Solution

This can only be the case if [tex]Y = \mathbb{R}^n \setminus \emptyset = \mathbb{R}^n[/tex] wouldn't it?

Cheers

Frank