Interior points proof where one set is a subset of the other

In summary, to prove that if A is a subset of B, the interior points of A are a subset of the interior points of B, we can use the definition of an interior point and show that if a is an interior point of A, then it is also an interior point of B since A is a subset of B. This can be proved by considering an open set Q in A containing a, and showing that because A is a subset of B, a and Q are also in B.
  • #1
ppy
64
0
How would I go about proving that if A is a subset of B then the interior points of A are a subset of the interior points of B?
 
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  • #2
What did you try?? Where are you stuck? What is your definition of interior point?
 
  • #3
If a is an interior point of A, the there is an open set Q in A containing a. Since A is a subset of B, a and Q are in B. Therefore a is an interior point of B.
 
  • #4
mathman said:
If a is an interior point of A, the there is an open set Q in A containing a. Since A is a subset of B, a and Q are in B. Therefore a is an interior point of B.

It would have been so much better if you let the OP find this himself.
 
  • #5


To prove that the interior points of A are a subset of the interior points of B, we can use the definition of interior points. An interior point of a set is a point that has an open neighborhood entirely contained within the set.

First, we can assume that A is a subset of B, meaning that every element in A is also in B. This means that any open neighborhood of a point in A is also a subset of B.

Next, we can take an arbitrary interior point x of A. By definition, there exists an open neighborhood N(x) of x that is entirely contained within A. Since A is a subset of B, this open neighborhood N(x) is also a subset of B.

Therefore, x is an interior point of B since it has an open neighborhood N(x) that is entirely contained within B. This holds true for any arbitrary interior point of A, meaning that all interior points of A are also interior points of B.

In conclusion, if A is a subset of B, then the interior points of A are a subset of the interior points of B.
 

1. What is an interior point?

An interior point is a point that lies within a set or a region. It is a point that is not on the boundary of the set and can be surrounded by other points within the set.

2. Can one set be a subset of another set?

Yes, one set can be a subset of another set. A subset is a set that contains elements that are also present in another set. In other words, all the elements of the subset are also present in the larger set.

3. How do you prove that one set is a subset of the other set using interior points?

To prove that one set is a subset of the other set using interior points, we need to show that every element in the first set is also an element of the second set. This can be done by showing that every interior point of the first set is also an interior point of the second set.

4. What is the significance of using interior points in a proof?

Using interior points in a proof allows us to show that one set is a subset of another set without having to consider all the points in the sets. It simplifies the proof by focusing on the interior points, which are easier to work with and can provide enough evidence to show the subset relationship.

5. Can we use interior points to prove that two sets are equal?

Yes, we can use interior points to prove that two sets are equal. If we can show that every interior point of one set is also an interior point of the other set, and vice versa, then we can conclude that the two sets are equal.

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