Question

Show that \(\bigcup_{n=2}^{\infty}\left[\frac{1}{n}, 1-\frac{1}{n}\right]\) is open in \(\mathbb{R}\).

Short answer

Answer: No, it is not open in \(\mathbb{R}\). However, its interior, \(\bigcup_{n=2}^{\infty}\left(\frac{1}{n}, 1-\frac{1}{n}\right)\), is open.

Step-by-step solution

Understand the sets

The given set is the union of an infinite number of sets \(\left[\frac{1}{n}, 1-\frac{1}{n}\right]\), where \(n\) ranges from \(2\) to \(\infty\). Each individual set contains all real numbers between \(\frac{1}{n}\) (inclusive) and \(1-\frac{1}{n}\) (inclusive).

Prove that each set is open

To prove that a set is open, we need to show that for every point \(x\) in the set, there exists an open interval \((x-\epsilon, x+\epsilon)\), where \(\epsilon>0\), which is completely contained in the set. However, since \(\frac{1}{n}\) and \(1-\frac{1}{n}\) are endpoints of the intervals, it's actually a closed set. To show that the union of these sets is open, we should consider their interior. The interior of a set is the largest open subset that can be formed by removing the boundary points. In this case, the interior of the set \(\left[\frac{1}{n}, 1-\frac{1}{n}\right]\) is the open interval \(\left(\frac{1}{n}, 1-\frac{1}{n}\right)\), which is obviously open.

Prove that the union of open sets is open

Now we will use the property that the union of open sets is also open. We have already established that the interior of each set \(\left[\frac{1}{n}, 1-\frac{1}{n}\right]\) is the open set \(\left(\frac{1}{n}, 1-\frac{1}{n}\right)\). Therefore, our desired set can be rewritten as \(\bigcup_{n=2}^{\infty}\left(\frac{1}{n}, 1-\frac{1}{n}\right)\), which is a union of open sets. For any point \(x\) in the union of these sets, there exists an open interval \((x-\epsilon, x+\epsilon)\), where \(\epsilon>0\), which is completely contained in one of the sets \(\left(\frac{1}{n}, 1-\frac{1}{n}\right)\). Since this is true for all points in the union, we can conclude that the set \(\bigcup_{n=2}^{\infty}\left(\frac{1}{n}, 1-\frac{1}{n}\right)\) is also open. Thus, we have shown that the set \(\bigcup_{n=2}^{\infty}\left[\frac{1}{n}, 1-\frac{1}{n}\right]\) is open in \(\mathbb{R}\).

Key concepts

Open Sets

In topology, open sets are one of the foundational elements. An open set is a set where, for any point inside it, there's a small "wiggle room" interval entirely within the set. Imagine walking inside a train carriage; you should be able to take a step in any direction without bumping into a wall, symbolizing the boundary of the open set.

In mathematical terms, a set is open if, for every point \(x\) within the set, there exists a positive distance \(\epsilon\), such that all points within \(\epsilon\) distance from \(x\) are also in the set. This definition is crucial because it guarantees the absence of boundary points within the set, differentiating it from closed sets which include their boundary. For instance, the open interval \((a, b)\) in \(\mathbb{R}\) includes all points greater than \(a\) and less than \(b\), but not the endpoints \(a\) and \(b\) themselves.

Recognizing open sets allows us to better understand the structure of spaces in mathematics, facilitating further exploration and analysis.

Interiors

The interior of a set provides the largest open set that can fit entirely inside the original set. Picture this like the inner room of a house, where you are assured warmth without stepping into chilly outer rooms.

This concept of the interior is key in topology, helping to dissect sets and understand their openness. If you have a closed interval \([a, b]\), its interior would be the open interval \((a, b)\), excluding the endpoints. Importantly, the interior is naturally open. In our exercise, each original closed set \(\left[\frac{1}{n}, 1-\frac{1}{n}\right]\) has an interior \(\left(\frac{1}{n}, 1-\frac{1}{n}\right)\) that is open, proving helpful in showing the broader openness of their union.

Finding the interior helps in visualizing the heart of a set, cutting away the boundaries and revealing the true open space within. It's a powerful tool, simplifying complex structures into open, manageable pieces.

Union of Sets

The union of sets is equivalent to gathering all elements from multiple sets into one larger set. Imagine collecting all favorite books from different friends into one big bookshelf. Each book matters and adds to the collection, making it richer and fuller.

In topology, a precise property is that if each set in a collection is open, then their union is open too. This concept was pivotal in our exercise, where the union \(\bigcup_{n=2}^{\infty}\left(\frac{1}{n}, 1-\frac{1}{n}\right)\) was proven open since each component set was open. Using this property helps simplify and solve complex problems by focusing on the openness of individual component sets.

Understanding unions allows mathematicians to explore vast spaces by pooling together smaller, manageable parts. It's a fundamental operation enabling deeper dives into more extensive topological structures and their properties.