Here you will cover interior and non-interior points by defining them and finding out how they manipulate other sets. You will also draw a couple of pictures that will tell you whether they are interior or non-interior points. If the events are in the set, they are part of the mint actions. To be a non-interior point, it has to be a point that is not in the set. You will see how you can prove that a point is a non-interior point by considering the intervals. You will understand how to find nint and int values as you do the necessary calculations for intervals. There are also exercises to fill in the details of calculations done. The rationals have no interior points.
Lecture 6: Math. Analysis – interior points/ non-interior points
Now that you know the interior and the non-interior it is easy for you to move straight to open sets. It is great for you when dealing with complex functions that have two dimensions only. Open means that all its points are interior points. To be open if the interior is a whole view and all the points are interior views. However, open does not mean that it is empty. The non-interior points are at the end of an interval. You also need to use these intervals to prove whether the values are open sets or not. You can use the triangle law to find these proofs.
Lecture 7: Math. Analysis – Topology of d-dimensional Euclidian space
Chapter for covers the topology of Rd. you will learn the differences between closed and not open which most people think are the same thing. It starts with a lemma that says if you make a set bigger, you cannot make the interiors smaller. In this video, the professor strives to prove this. Ensure that the values you choose in your calculations do not allow for infinity values. You will also learn different proof methods. You will also understand unions and how they work in the sequences on open sets. You will also find out which intersections are open and which ones are not.
Lecture 8a: Math. Analysis – Closed sets
It continues the lessons learnt from chapter 4. You can use the induction to prove that if the intersection of one, two and three are open, the rest of them should be open. With closed sets, closed does not mean not open. Open sets are critical for complex functions, but they are not prerequisites. The more information there is in your diagram, the better. From the diagrams you create, people can see what you are trying to argue and how you have gone about proving what you think is true and what you think is false. You also learn how to look at complements to find out whether sets are open or closed.
Lecture 8b: Math. Analysis – Sequences in d-dimensional Euclidian Space
This is a brief introduction of chapter five where you will revisit the notion of convergent sequences, and the setting will be extended to Rd. Convergent sequences are difficult to prove. Some of the concepts discussed here will also be expanded on in chapter nine. It will also help you practice more with proofs before you get to chapter nine. Sequences may also contain repeated terms. You will also have to use integers so it may be common for you to eliminate natural numbers. The video also contains a secret as to why students lose a lot of marks during the exams when calculating issues on convergence.