Open Sets in the Plane
The notion of open sets is a way of formalizing and generalizing the notion of closeness in the plane; that is, it is a way to formally talk about points in the plane whose distance apart is very small, according to some distance function. Up until now, we have looked at each notion of distance we have encountered separately. Talking about open sets is a way to talk about all the distance functions we have encountered at the same time.
Let's start by defining what an open set is. When we talk about open sets, we always have a particular distance function in mind. So let's fix a distance function on the plane, say d(x, y). A subset of the plane is open if for every point (x, y) in the subset, there is some number positive number a(x, y) so that all points within distance a(x, y) of (x, y) are also in the subset.
The entire plane is an open subset, as is the empty set (the set with no points in it). (Why?)
Problem 1: Consider the Euclidean distance function. Is the subset of the plane consisting of all the points less than distance 1 from the origin open? What about the points of distance exactly 1? What about the points of distance greater than 1?
If two different distance functions give the same open sets in the plane, then points are "close" with respect to one distance function if and only if they are "close" with respect to the other. The next problem will give you a more formal idea of what this means.
Problem 2: If a subset is open with respect to Euclidean distance, is it also open with respect to cab driver distance? What if a subset is open with respect to cab driver distance, is it also open with respect to Euclidean distance?
If two points are very close together with respect to a given distance function, then there is some open set containing them that is not equal to the entire plane. (Why is this true?) So, if we are able to say something about all the open sets with respect to a particular distance function, then we are also saying something about points that are close together with respect to that distance function. If we can say something about open sets in general, for an arbitrary distance function, then we are saying something about the property of "being close" to another point, regardless of the distance measure we use.
Problem 3: Is the intersection of two open sets always open? Why or why not? Intuitively, what does this say about points that are close to both point A and point B? If points S and T are both within distance r of both points A and B, what is the farthest apart points S and T can be?
Problem 4: Is the union of two open sets open?
A function f that takes points in the plane to points in the plane is called continuous if, given an open subset of the plane, say O, the subset of all the points (x, y) in the plane such that f(x, y) is in O is also open.
Problem 4: If two points in the plane are very close together, what can you say about their images under a continuous function?
Problem 5: Is the composition of two continuous functions that take points in the plane to points in the plane again a continuous function? What does this say about applying a continuous function to points that are close together, and then applying another continuous function to the result?
Problem 6: Consider the function f(x, y) = (x, y) from the plane to the plane. For points in the domain consider the Euclidean distance function. For points in the image consider the cab driver distance function. Is f continuous? What if we considered the cab driver distance function in the domain, and the Euclidean distance function in the image?
Problem 7: Are all constant functions from the plane to the plane continuous? Why or why not?
A subset of the plane is called closed if its complement is open. So, being able to talk about closed sets lets us talk about points that are "far away" from a given point.
Problem 8: Say whether the following subsets of the plane are open, closed, or neither with respect to the Euclidean distance function. Explain your reasoning.
1. The points distance exactly 1 away from the origin.
2. The points distance exactly 1 away from the origin together with the origin.
3. The points with strictly positive y coordinate, together with the origin.
4. The points with strictly negative x coordinate.
Problem 9: Is the intersection of two closed sets closed? What about the union of two closed sets? Intuitively, what does this say about the points that are far away from either of two points that are close together?