An infinite union of closed sets is a collection of sets that includes an unlimited number of closed sets. It is considered to be "not closed" when the union of all these sets does not have a closed boundary, meaning that there is at least one point that does not belong to any of the sets in the union.
An example of an infinite union of closed sets that isn't closed is the union of all closed intervals [0, 1/n] for n = 1, 2, 3, ..., infinity. This union includes an infinite number of closed sets, but the point 0 is not included in any of the intervals, making the boundary not closed.
An infinite union of closed sets that isn't closed is significant in topology and analysis because it challenges the intuition that the union of closed sets should also be closed. This concept highlights the importance of understanding the boundary of a set and how it can affect the properties of the set.
Yes, an infinite union of closed sets can be closed in certain scenarios. For example, if the sets in the union have a finite intersection, then the union will also have a closed boundary and therefore be closed.
An infinite union of closed sets that isn't closed is an example of a non-compact set. This means that the set cannot be covered by a finite number of open sets, making it a counterexample to the concept of compactness. This highlights the importance of understanding the limitations of compact sets and the need for more general definitions, such as that of a closed set.