Is the Translation of Open and Closed Sets in R^n Also Open or Closed?

In summary, the conversation discusses how to show that a subset A of Rn is open or closed depending on the translation by a point vector \vec{w}. The solution involves using the componentwise approach and brute forcing the definition of open.
  • #1
Appa
15
0

Homework Statement


Let A be a subset of Rn and let [tex]\vec{w}[/tex] be a point in Rn. Show that A is open if and only if A + [tex]\vec{w}[/tex] is open.
Show that A is closed if and only if A + [tex]\vec{w}[/tex] is closed.

Homework Equations



The translate of A by [tex]\vec{w}[/tex] is defined by
A + [tex]\vec{w}[/tex] := {[tex]\vec{w}[/tex] + [tex]\vec{u}[/tex] | [tex]\vec{u}[/tex] in A}

The Attempt at a Solution


I tried to solve this componentwise:
[tex]\vec{u}[/tex] = {pi(ui)}, 1<=i<=n, so that [tex]\vec{u}[/tex] + [tex]\vec{w}[/tex] = {pi(ui) +pi(ui)}
But I'm not all that sure whether I'm on the right track..!
 
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  • #2
You can just brute force the definition of open: If you have a point u in A, you have a small ball around it contained in A, and that ball will be translated into A+w. You should be able to see how that gives you that A+w is open
 

Related to Is the Translation of Open and Closed Sets in R^n Also Open or Closed?

1. What is an open set in R^n?

An open set in R^n is a set of points that does not include its boundary. This means that for any point within the set, there is a small neighborhood around it that is also contained within the set.

2. How is an open set different from a closed set in R^n?

A closed set in R^n is a set of points that includes its boundary. This means that for any point on the boundary, there is no neighborhood around it that is entirely contained within the set. Unlike open sets, closed sets may also include points on the boundary.

3. Can a set be both open and closed in R^n?

Yes, in some cases a set can be both open and closed in R^n. This occurs when the set is the entire space, or when the set is empty. In these cases, the set contains all of its boundary points and therefore is both open and closed.

4. How are open and closed sets related to continuity?

In mathematics, continuity is a property that describes how a function behaves at points within its domain. In R^n, a function is continuous if and only if the pre-image of any open set is also open. This means that open sets are closely related to continuity, as they are used to define this property.

5. Are open and closed sets unique in R^n?

No, there are many different open and closed sets in R^n. For example, the set of all points with a distance less than 1 from the origin is an open set in R^2, but so is the set of all points with a distance less than 2 from the origin. Similarly, the set of all points with a distance less than or equal to 1 from the origin is a closed set in R^2, but so is the set of all points with a distance less than or equal to 2 from the origin. There are infinitely many possible open and closed sets in R^n, each with different properties and characteristics.

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