The two sets are subsets of the Euclidian space .
I need to show that
(1) is open.
(2) is open.
I know according to the definition that a subset of is said to be open, if there for every point exist a real number given every point . Then and
(1) and (2)
Then if both are open subsets of by definition above. Then if there exist a point x and y in both subsets, then if there still exist a epsilon > 0, then their respective Union and Intersection is also open according to the definition above.
How does that sound? Or do I need to above something more concrete?
Or do I need to add something about the union of two open sets are closed? Bu if the union of two subsets are closed, then how possible can it be open?
I'm going to start by nitpicking your definition. I would have said something like this instead:
I realize that's what you meant, but it's important to be precise when defining stuff.A subset of is said to be open if there for every point exists a real number such that every point that satisfies |x - y| < is a member of .
I'll do 2 for you. You can probably handle 1 after that. I'm going to use the concept "open ball", so I'll define that first:
An open ball of radius r around a point x is the set of points y such that |x-y|<r.
Using this concept, we can state the definition of openness like this: A set V is open if, for every point x in V, there's an open ball around x that's a subset of V.
OK, this is the solution of 2:
Let x be a point in the intersection of V1 and V2. Then x is in V1 and x is in V2. Since x is in V1, there's an open ball around x that's a subset of V1. Since x is in V2, there's an open ball around x that's a subset of V2. The smaller of these open balls is a subset of the larger, and must therefore be a subset of both V1 and V2. That's equivalent to saying that the smaller of these two open balls around x is a subset of the intersection of V1 and V2, and that's exactly what we need to conclude that the intersection of V1 and V2 is open.
Edit: I see that Plato beat me to it. His solution is exactly the same as mine.
A union of open sets (even infinitely many) is always open. The intersection of a finite number of open sets is always open.Or do I need to add something about the union of two open sets are closed? Bu if the union of two subsets are closed, then how possible can it be open?