[MUSIC] What about intersection of open sets? Let us have a finite number of open sets. They're all open. Now, I form the intersection, that'll be A, using intersection symbol. It can be proven A is an open set. Just compare here we intersected finite number of open sets. In the previous case, we could have infinitely many open sets and we united them and as a result we've got an open set. It's not possible to intersect infinitely many open sets, because as a result we get a set which may be not open. For instance, let's consider the set Ai. Which is an interval- On the number line. Where i is a natural number. And now we form this intersection. Here i goes to infinity, so here I write infinity symbol. What do I get then? May be the result will be clearer if I draw the number line. So this is my origin, 0 point. And I have a sequence of open intervals, which are becoming smaller and smaller- With each i. And I have to intersect them. So what do I get? I get just one point 0 point. But threat containing just one number, just one point is not open. So that's why we restrict ourselves with the finite number of open sets. Now, we proceed with the sequences. Sequences- In Rn. It's clear what a sequence is in R. Now, if I choose- X chrome Rn and I need to form a sequence, it's important to label them. Because the subscript over the letter X indicates a coordinator, I have to use an upper script. I will be using m within brackets. M is an initial number. When I draw the braces And here I put m. M ranges from 1 to infinity. This is the sequence. That is how we denote it. IN or Rn. But if I choose particular m. For example I write x3- I call it, this is the third term or the sequence. 3rd term. We know from single variable calculus what is a convergent sequence. A convergent sequence is a sequence which has a limit. Here in Rn, we also can write similarly- When m tends to infinity, the sequence has a limit which equals x, let's suppose x0. But this is just a notation expressing the fact that to this is a convergent sequence whose limit is x0. But we need to find a proper definition, will base our definition on the notion of a distance in a dimensional space. So here I'll write down the definition, what it means that x0 is the limit of the sequence. So I'll be using symbols. For any epsilon greater than 0. There exists- M depending on epsilon, such that for all m greater or equal to m epsilon, The following inequality holds the distance between the m's term or this sequence, and the limiting points- Should be less than epsilon. So this is a definition or the fact that this is a convergent sequence whose limit is x0. Now, we are approaching the concept of a closed set in Rn. So the next topic will be closed- Closed sets. Before starting with rigorous definitions- Let me provide some intuition into this concept. We would like to have a property of a set, such that given any convergence sequence whose terms belong to the set. We would like to be sure that its limit also belongs to the same set. Now, let me write down a set, S From Rn- Is called closed, or is closed. If- Any convergent sequence. Sequence. Who's elements belong to this set- Has the limit- Which belongs to S. [MUSIC]