Is the Intersection of Open Sets Always Open?

In summary, we need to prove that for any collection {Oα} of open subsets of ℝ, \bigcap Oα is open. Consider an arbitrary x in O= \bigcap Oα and by definition x is in O, and O is open by hypothesis. So x is an interior point of Oα. However, it is unclear how to proceed with the intersection of a set, as shown by the example of ##O_n = (-\frac 1 n, \frac 1 n)##.
  • #1
mathanon
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Prove that for any collection {Oα} of open subsets of ℝ, [itex]\bigcap[/itex] Oα is open.


I did the following for the union, but I don't see where to go with the intersection of a set.

Here's what I have so far:

Suppose Oα is an open set for each x [itex]\ni[/itex] A. Let O= [itex]\bigcap[/itex] Oα. Consider an arbitrary x in O. By definition of O, x is in O, and O is open by hypothesis. So x is an interior point of Oα
 
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  • #2
mathanon said:
Prove that for any collection {Oα} of open subsets of ℝ, [itex]\bigcap[/itex] Oα is open.

What if ##O_n = (-\frac 1 n, \frac 1 n)##?
 

FAQ: Is the Intersection of Open Sets Always Open?

What is an open set in mathematics?

An open set in mathematics is a set that does not contain its boundary points. In other words, for any point in an open set, there exists a small enough neighborhood around that point that is entirely contained within the set. This means that there are no points on the edge or boundary of the set that are not also included in the set.

How is an open set different from a closed set?

A closed set is the complement of an open set, meaning it contains all of its boundary points. This means that there are no points on the edge or boundary of the set that are not included in the set itself. An open set and a closed set are mutually exclusive, meaning a set cannot be both open and closed at the same time.

What is the importance of open and closed sets in topology?

Open and closed sets are fundamental concepts in topology, as they help define the properties and structure of a topological space. These sets are used to define continuity, connectedness, and compactness, among other important topological properties. Understanding open and closed sets is crucial for studying and analyzing topological spaces.

Can a set be both open and closed?

No, a set cannot be both open and closed. As mentioned earlier, open and closed sets are mutually exclusive. A set can be either open, closed, or neither. However, in some cases, a set may be both open and closed in different topological spaces. For example, in the discrete topology, all sets are both open and closed.

How are open and closed sets related to the concept of limit points?

Limit points are points that are arbitrarily close to a given set. In topology, limit points are often used to define open and closed sets. For example, for a set to be open, every point in the set must have a neighborhood containing only points in the set. This means that there are no limit points outside of the set. On the other hand, for a set to be closed, it must contain all of its limit points.

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