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poutsos.A
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How do we prove that the following set is open??
S= [tex]R^2[/tex]- {(x,y) : y=[tex]x^2[/tex], xεR}
WHERE R is for real Nos
S= [tex]R^2[/tex]- {(x,y) : y=[tex]x^2[/tex], xεR}
WHERE R is for real Nos
poutsos.A said:Thanx, but how do we calculate that distance d(x,y) =r??
As long as you know there is such a non-zero distance, you don't need to know what it is!poutsos.A said:Thanx, but how do we calculate that distance d(x,y) =r??
To prove the openness of a set in R^2 means to show that the set contains all of its interior points. In other words, every point in the set has a neighborhood contained entirely within the set.
To prove the openness of a set in R^2, you can use the definition of openness, which states that a set is open if every point in the set has a neighborhood contained within the set. This can be shown by using the distance formula to find the distance between a point in the set and all other points in the set, and then finding a radius that is smaller than the distance to ensure that all points within that radius are also in the set.
Proving the openness of a set in R^2 is important because it allows us to determine whether a set is continuous or not. In other words, if a set is open, we can be sure that it does not contain any boundary points, which are points that are both in and not in the set. This is a crucial concept in various areas of mathematics, including calculus and topology.
No, a set cannot be both open and closed in R^2. A set is considered open if it contains all of its interior points, while a set is closed if it contains all of its boundary points. Since a set cannot contain both interior and boundary points, it cannot be both open and closed.
Yes, there are alternative methods for proving the openness of a set in R^2, such as using the definition of openness in terms of limit points or using the concept of open balls. These methods can be useful in different scenarios and can provide alternative perspectives on the openness of a set.