CantorSet said:Hi everyone,
I'm reading Rudin's Analysis and in the topology section, he implies that the finite intersection of closed sets is not necessarily closed. (pg. 34)
Can someone give an example of this? I can't seem to find one.
A closed set is a set that contains all of its limit points. In other words, every convergent sequence in the set converges to a point within the set.
Yes, consider the closed sets [0,1] and [1,2] in the real numbers. The intersection of these two sets is the single point {1}, which is not a closed set since it does not contain all of its limit points (in this case, the limit point 1 is not in the set).
This is because the limit points of the intersection may not be contained in the intersection itself. In other words, the intersection of closed sets may not have all of its limit points, making it not closed.
No, the converse is not necessarily true. While a finite intersection of open sets is often open, there are cases where it is not. For example, consider the open sets (-1,0) and (0,1) in the real numbers. The intersection of these two sets is the single point {0}, which is not open.
In topological spaces, the concept of closed and open sets is generalized. While in Euclidean spaces (like the real numbers), closed sets are defined as sets that contain all of their limit points, in topological spaces, closed sets are defined as sets whose complement is open. The idea of finite intersection of closed sets not necessarily being closed is a result of this more general definition of closed sets in topological spaces.