Infinite Unions of Open/Closed Sets: Explained

[aaaa202]

If you unite infinitely many open sets you still get an open set whilst the same is not necessarily true for a closed set. Can someone try to explain what property of a union of open sets it is, that assures that an infinite union is still open (and what property is the closed sets missing?)

 

[jedishrfu]

While not directly related to your question, wikipedia discusses the clopen set:

http://en.wikipedia.org/wiki/Clopen_set

They mention that a set may be both open and closed mathematically, that the definitions of open and closed are not mutually excluse and provide examples.

Perhaps from this you can answer your question.

 

[Fredrik]

In the context of topological spaces, it's a definition (of "topology" and "topological space"), so it doesn't require an explanation. In the context of metric spaces, it's easy to prove, but the details depend on what definition of "open" you're using. One very common definition says that a set is open if and only if all its elements are interior points. I suggest that you use this definition to prove it yourself. You can start the proof like this:

Let $\{E_i:i\in I\}$ be an arbitrary sequence of open sets. Let $x\in\bigcup_{i\in I}E_i$ be arbitrary.

Now you just need to show that x is an interior point of $\bigcup_{i\in I}E_i$.

For closed sets, you just need a counterexample. Consider e.g. the intervals [0+1/n,2-1/n] where n is a positive integer. What is the union of all of them?

 

[mathman]

If you unite infinitely many open sets you still get an open set whilst the same is not necessarily true for a closed set. Can someone try to explain what property of a union of open sets it is, that assures that an infinite union is still open (and what property is the closed sets missing?)

To some extent it depends on what you are starting with. In the abstract the open sets are defined to have the properties, closed under all unions and finite intersections. Closed sets are then defined as complements of open sets, and therefore closed under finite unions and all intersections.

 

[blue_raver22]

The property that guarantees that an infinite union of open sets is still open is called closure under arbitrary union. This means that for any collection of open sets, their union will still be an open set. This property is a fundamental characteristic of topological spaces, which are mathematical structures used to study the properties of sets and their subsets.

On the other hand, closed sets do not necessarily have this property. This means that an infinite union of closed sets may not necessarily be a closed set. This is because the definition of a closed set is that it contains all of its limit points (points that can be approached arbitrarily closely by points in the set). However, when taking an infinite union of closed sets, there may be limit points that are not contained in the union, making it not closed.

In other words, the property that open sets have and closed sets lack is closure under arbitrary union, which ensures that the union of infinitely many open sets will still be an open set. This is a crucial concept in topology and is used to prove many important theorems in mathematics.