Proving Closure of a Simple Set

In summary, the conversation discusses different approaches to proving that a simple set, specifically the set of points defined by the parabola y=x^2, is closed. Suggestions include using the theorem that states if f: X -> Y is a continuous function and V is contained in Y (where V is closed), then f^-1(V) is closed, or showing that the complement of the set is open by finding the distance from a point outside the set to the set and showing that if the distance is 0, the point is in the set. Another suggestion is to find a function that equals 0 if and only if y=x^2, although proving that the function is an embedding does not necessarily imply that the set is closed
  • #1
dracox
5
0

Homework Statement


How do I show that a simple set is closed?

Ex: the set of points defined by the parabola y=x^2



The Attempt at a Solution


Well, a set is closed iff it contains all of its limit points. So, I want to show that this is true for the given set. I'm not exactly sure how to do this. Any help?
 
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  • #2
It's often easier to use certain theorems to prove things like this than trying to prove them from the definition.

Couldn't you use the theorem that states that if f: X -> Y is a continuous function and V is contained in Y (where V is closed) then f^-1 (V) is closed?

EDIT: nevermind. I didn't see the example you provided in the OP. This theorem won't work.
 
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  • #3
Hmm, I'm not familiar with this theorem. Is there somewhere I could read about it on the web? Or, if you're inclined, you could explain it. What would V be in this case?
 
  • #4
Ah, nevermind, I am familiar with this theorem. I just found it in my textbook. How shall I apply it, though?
 
  • #5
In the example case it is easiest to say that there is a homomorphism from a curve to R, since being closed (or open) is a topological property.

Another way should be to show that its complement is open by taking a point outside, finding its (least) distance to the set and then showing that if that distance is 0, the point is in the set.

I'm not sure how to apply the definition directly.
 
  • #6
Ok, I think I'll try it your second way. Let (x_0, y_0) be a point in the complement of the set (call it A). How would I proceed from here?
 
  • #7
Anyone?
 
  • #8
Well, since you have a curve, it is easy to write a function of distance from some point to (x0, y0). Then the least distance d(x, y) is the minimum of that function. Then show that d(x, y) = 0 means that y = x^2.

Although proving that x -> (x, x^2) is an embedding is a lot more understandable and straightforward.
 
  • #9
Can you find a function [itex]f:\mathbb{R}^2\rightarrow \mathbb{R}[/itex] such that f(x,y)=0 if and only if y=x²??
 
  • #10
hamsterman said:
Although proving that x -> (x, x^2) is an embedding is a lot more understandable and straightforward.

But being an embedding does not immediately imply closed.
 
  • #11
It should, if it's an embedding from R, I think.
 
  • #12
hamsterman said:
It should, if it's an embedding from R, I think.

We have that

[tex]\mathbb{R}\rightarrow \mathbb{R}^2:x\rightarrow (e^x,\sin(\frac{1}{e^x}))[/tex]

is an embedding that is not closed.
 

What is meant by "proving closure" of a simple set?

Proving closure of a simple set means showing that for any operation on elements in the set, the result will still be an element in the set. In other words, the set is "closed" under the given operation.

Why is proving closure important in mathematics?

Proving closure is important because it allows us to make conclusions about a set without having to explicitly check every element. It also helps us to understand the behavior of operations on the set and make predictions about the set's elements.

What are some typical operations used to prove closure?

Common operations used to prove closure include addition, subtraction, multiplication, division, exponentiation, and taking limits or derivatives. The specific operation depends on the type of set being considered.

How is proving closure different from proving convergence?

Proving closure is related to showing that an operation on elements in a set will always produce an element in the same set. Proving convergence, on the other hand, is related to showing that a sequence of elements in a set will approach a certain limit or value. Both are important concepts in mathematics, but they have different focuses and techniques.

What are some strategies for proving closure of a simple set?

One strategy for proving closure is to use algebraic manipulations to show that the result of the operation on two elements in the set is still an element in the set. Another strategy is to use logical arguments, such as mathematical induction, to prove that the set is closed under the given operation. Additionally, counterexamples can be used to disprove closure, so it is important to consider potential exceptions when proving closure.

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