When we discussed open sets, I provided example of an open set. That was the first quadrant. It's open but it's not closed. Why is that? Because we can easily find a sequence. Let me draw the picture, like this. It's coming closer and closer to the horizontal axis X1. Let it be a sequence, where the first coordinate is simply one, and this is one. Here, we have one over m, right? So, what about the limit or the sequence? So, all elements or terms or the sequence, they belong to this set. But as a result, we have one, zero, but this point doesn't belong to R, R2 plus. We have discussed, this is open but it's not closed. By the way, when we look at this sequence, we found easily it's limits because we considered separately all these coordinates are sequences themselves. So, this sequence is a convergent sequence whose limit is zero. There is a general fact, that whenever we have a sequence in n-dimensional space, whose limit is zero-. So, if we have this, then for each coordinate i, the limit exists and it equals ith of coordinate of the limit, and the same is true in the opposite direction. So, whenever each coordinate sequence is convergent and has some limit, then sequence in Rn is also convergent, and its limit consists of limiting coordinates. Now, let's consider what operations can be perform on the closed sets. It can be proven that whenever we intersect closed sets, so, A alpha is a closed set. Alpha is an index, which ranges within some set i. The intersection is also a closed set. What about the union? It can be proven that when we unite a finite number of closed sets, this time, i ranges from one to sum P, this is also closed set. But in the same fashion as we have seen with the open sets, when we try to unite infinitely many sets, we get not necessarily a closed set. Bounded sets. A definition will follow. A set S from n-dimensional space Rn is called bounded if there exists a ball of radius R centered at zero such that S belongs to this ball, and the final definition, a compact set. A set S from Rn which is closed and bounded is called compact. Let me provide some examples. A closed ball, a closed ball is a compact. How did we know it is? Because we were using capital B, but that was for the open set; now we have a closed set. The difference is that, in the definition of a closed ball, we use non-strict inequality. So, I'm taking the boundary points or this ball and I add them. This operation is called closure. So, I'll be using Cl, which means closure. Closure of the ball centered at x0 is the closed ball, and that's the definition. That's the distance between x and the center. Here, non-strict inequality. So, that's the definition of the closed ball and I'm indicating these by using Cl, which means closure. This is an example of a compact set. First of all, it's closed, and it can be proven. Also, it's bounded, because we can always find a ball with a greater radius which includes this one. On the number line, the compacts are usually closed intervals or it can be a union of some closed intervals. There is a theorem from the single variable calculus, which is known by the name of its author, Weierstrass theorem which claims that a continuous function on a closed interval takes its greatest and least values, and this is a very important theorem which is widely used in calculus itself and its applications, and we'll see later that there is an analog of this theorem in multivariate calculus. We will be considering continuous functions on compacts. So analogue of a closed interval in n-dimensional space is a compact.