91Ó°ÊÓ

An open set is a collection of points where, for any point within the set, there exists a neighborhood completely contained within the set. In simple terms, you can "wiggle around" any point and still stay inside the set. In the context of real numbers, a basic example of an open set could be an open interval \((a, b)\) where neither endpoint \(a\) nor \(b\) is included. What's important here is that no matter where you pick a point in this interval, you can always find a small range around it that stays fully inside the interval.

For the set of rational numbers \(\mathbb{Q}\), being dense in \(\mathbb{R}\) means between any two real numbers, there are rational numbers. However, if we pick any rational number and try to find a neighborhood in \(\mathbb{Q}\) only, it won't fully exist due to the nearby irrational numbers. Therefore, \(\mathbb{Q}\) cannot be open because such neighborhoods can't be confined to only rational numbers.

A closed set, on the other hand, is a set that contains all its limit points. This means that if you ever find a point where a sequence of elements from the set seems to approach, that point must also be part of the set.

Closed sets include their boundary points. For example, the closed interval \([a, b]\) includes the numbers \(a\) and \(b\). This is opposite to open sets, where boundary points aren't included.

In the case of \(\mathbb{Q}\), there are many sequences of rational numbers converging to irrational numbers like \(\sqrt{2}\), but \(\sqrt{2}\) is not in \(\mathbb{Q}\). Hence, \(\mathbb{Q}\) doesn't include all its limit points, so it’s not closed.

Limit points, or accumulation points, are points that can be "approached" by elements of a set. This means you can find elements from the set as close to the limit point as you wish, even if the point doesn't belong to the set.

The concept of a limit point is crucial in discussing open and closed sets because it helps define whether a set is truly complete within its boundaries (closed) or not (open). A classic feature of rational numbers \(\mathbb{Q}\) is that it has limit points that are irrational, like \(\sqrt{2}\).

To see this, consider a sequence of rational numbers getting arbitrarily close to \(\sqrt{2}\). Even though \(\sqrt{2}\) is not rational, the sequence shows \(\sqrt{2}\) as a limit point of \(\mathbb{Q}\). It's precisely this property of having limit points outside the set that indicates \(\mathbb{Q}\) is neither open nor closed.