Question

Prove: (a) The intersection of finitely many open sets is open. (b) The union of finitely many closed sets is closed.

Short answer

Question: Prove the following statements: a) The intersection of finitely many open sets is open. b) The union of finitely many closed sets is closed. Answer: a) Given a finite collection of open sets, their intersection results in an open set, as every point in the intersection has a neighborhood completely contained in the set, satisfying the definition of an open set. b) Given a finite collection of closed sets, their union results in a closed set, as every limit point of the union is contained in the union, satisfying the definition of a closed set.

Step-by-step solution

Proof of (a): The Intersection of Finitely Many Open Sets is Open

Let's consider a finite collection of open sets, \(\{U_1, U_2, ..., U_n\}\). We need to prove that their intersection, \(U = \bigcap_{i=1}^n U_i\), is also an open set. Recall that a set is open if and only if every point in the set has a neighborhood completely contained in the set. So, let \(p\) be an arbitrary point in the intersection \(U\). Since \(p \in U\), it implies that \(p \in U_i\) for each \(i = 1, 2, ..., n\). As \(U_i\) are open, there exists an open ball \(B_i(p, r_i)\) around \(p\) such that \(B_i(p, r_i) \subset U_i\), where \(r_i > 0\). Let \(r = \min\{r_1, r_2, ..., r_n\}\). Now, we construct an open ball around \(p\) with radius \(r\), say \(B(p, r)\). We will show that this open ball \(B(p, r)\) is completely contained in the intersection \(U\). Let \(q \in B(p, r)\). Then, by the definition of the open ball, \(d(p, q) < r\). Now, since \(r \le r_i\) for each \(i\), we have \(d(p, q) < r_i\). This implies \(q \in B_i(p, r_i)\) for each \(i\). Thus, \(q \in U_i\) for all \(i\) and therefore \(q \in \bigcap_{i=1}^n U_i\). Since \(q\) was an arbitrary point in \(B(p, r)\) and it is in the intersection \(U\), we can conclude that \(B(p, r) \subset U\). Therefore, every point in \(U\) has a neighborhood entirely in \(U\), and by definition, \(U\) is an open set.

Proof of (b): The Union of Finitely Many Closed Sets is Closed

Let's consider a finite collection of closed sets \(\{C_1, C_2, ..., C_n\}\). We need to prove that their union, \(C = \bigcup_{i=1}^n C_i\), is also a closed set. Recall that a set is closed if and only if it contains all its limit points. We will show that every limit point of the union \(C\) is in \(C\). Let \(p\) be a limit point of the union \(C\). It implies that every neighborhood of \(p\) contains at least one point of \(C\). Since \(p\) is a limit point of \(C\), and \(C\) is the union of the sets \(C_i\), for each neighborhood of \(p\), there exists an \(i\) such that the neighborhood contains at least one point from \(C_i\). Now, since \(C_i\) is a closed set, it contains all its limit points. As \(p\) is a limit point of \(C_i\), it must be in \(C_i\). And as \(p \in C_i\), it must also be in the union \(C = \bigcup_{i=1}^n C_i\). Thus, every limit point of the union \(C\) is in \(C\). Therefore, the union \(C\) is a closed set.

Key concepts

Open Sets

Understanding open sets is fundamental in real analysis, and it hinges on grasping the intuitive idea of 'space' around points. An open set is essentially a collection of points that has this breathing room around every point within it. This 'breathing room' is represented by an open ball with a positive radius, which does not breach the set's boundaries.

Consider a simple example of an open interval on the real number line, such as \( (a, b) \), where every point in that interval has other points of the interval close to it without touching the interval's endpoints. In higher dimensions, this is akin to a sphere or a 'bubble' around each point that stays entirely within the set. The exercise you're working on demonstrates a crucial property of open sets: when you take the intersection of a finite number of open sets, the 'space' around each point remains intact, ensuring the intersection is also an open set.

This property of open sets is analogous to piling a bunch of sieves together; each having holes, their overlay, which represents the intersection, will still let water through—indicative of the openness of the intersection set.

Closed Sets

Closed sets, on the other hand, are on the opposite end of the spectrum in real analysis. These sets are characterized by their inclusivity of boundary points or 'limit points.' A set is regarded closed if it contains all the points that are arbitrarily close to points within the set—these are the limit points or the boundary points.

For instance, a closed interval \( [a, b] \) on the real line includes the endpoints a and b, representing its closed nature. Now, considering the union of closed sets as mentioned in your exercise, imagine drawing boundaries around each closed set and then merging them together. While the insides may or may not overlap, the resulting set will still have a clear boundary, effectively containing all limit points and thereby, by definition, being closed.

The implication of the exercise for closed sets is much like drawing different shapes on a paper and then shading over them all; you might cover some areas multiple times, but the outermost edges—no matter how irregular—still form a continuous boundary line of the shaded region.

Limit Points

Limit points, also known as accumulation points, are somewhat like the social connectors in a network—they are points that have other points from the set that can get arbitrarily close to them without actually being those points. It is essential to distinguish between points that are in the set and those that nudge infinitely close to the set; the latter can be outside the set in case of open sets or included in the set in the case of closed sets.

A more tangible way to envision limit points might be to think of the end point of a crowd gathering where there's no strict barrier; people can stand as close to the edge of the crowd as possible, representing limit points. In the context of your exercise, the importance of limit points manifests in understanding set closure: a closed set is precisely defined by its ability to embrace these sociable limit points, never leaving them isolated.

The articulation of limit points in the solution makes them the linchpin to deciphering why the union of closed sets remains closed—the sets contain not just the 'insiders' but also the closest 'outsiders,' making sure that every point on the cusp is accounted for.