Undergraduate Topology: Jan 20, definition of topology, examples

Here we begin a short course in Topology. We're following Marco Manetti's text with insights added from Munkres and other things sitting around my office. There are three students in this course and they are of course responsible for my mistakes and oversights.

Keep in mind, this course is aimed at these three students in particular. They have already worked with metric topology in other coursework and the big picture of topology was covered in their previous class. I also decided to cover background material as we come to it.

[fisikalectures]

FINALLY, I CAN UNDERSTAND WHAT THE PROFESSOR IS TALKING ABOUT!!!! I LOVE YOU

[AdenKhalil]

I've just started watching these lectures, and I 'm already liking'em

[taylorrendon]

I just have to say finding these lectures today actually made my day! Great handwriting and so far wonderful precision. I’m excited to learn from you.

[DonaldLancon]

26:45 - 36:52 Induction is not at all necessary here, since there is already a finite collection of open balls (around x) implied by the assumed finite collection of open sets. Thus, taking the minimum of the (finite number of) epsilons works just fine (i.e., that "minimum-epsilon" ball is contained within every other ball, and thus every Ui, and thus the intersection). Right?

[adityaprakash]

Are there lecture notes for this course uploaded anywhere?

[tmm]

wow! I'm impressed that you actually pray for your students. From my vantage point, it looks like you see your own children and really care about them.

[allyourcode]

@40:20 To give some motivation for this name, later, subspaces will almost certainly be introduced. Then, the following simple example can be considered: the integers induces a subspace of the reals. The induced topology of this subspace is the power set of the integers. Since we would informally the integers as "discrete", the formal definition of "discrete" seems quite fitting (at least in this case).

Another way to think about it is that {x} is "open" in the discrete topology. Therefore, every x in X lives in an "isolated neighborhood"; in other words, every point is separate from all other points ("separate" is a term that will likely be defined formally later, but for the time being, the informal/intuitive notion is not misleading and therefore suffices).

A small thm that I just thought of (proof is left as an exercise to the reader): the discrete topology is the only topology where every x in X is "isolated" (i.e. {x} is in the topology). This gives further justification to the name "discrete".

PS: BTW, it looks like this course doesn't mention it, but there is an alternate (and equivalent, but less often used) definition of a topology that uses "neighborhoods" instead of "open sets". The open sets formulation is almost always used, so that's probably why neighborhoods is not mentioned.

PPS: I guess the advantage of the neighborhoods definition is that it is easy to visualize: a neighborhood of a "point" is an "area" "around" the point. By following this approach, it becomes clear why open sets are special: an open set is a neighborhood of all of its elements (and this is not true of non-open neighborhoods).

[sophiemcgregor]

I am a graduate student in Mathematics... And i want to focus on analytic number theory and then go to algebraic geometry... this class was very nice.. i've been watching your lecture videos since i decided to pursue higher mathematics.. do you have any suggestions for me that i can do to enhance my skills???

[kiwanoish]

Really big thanks for making all these videos availble. And, I guess the praying always felt a bit awkward and weird, not annoying, but it honestly feels even less so after your words in the beginning of this video =). Just shows humor and distance, and makes me relate ;)

[UnforsakenXII]

Only three students? Wow. I've never seen such a small class.

[YungassPadawan]

I've always had a hard time with this kind of math. The definition he gives after "Metric Space" is so hard for me to wrap my head around.

[NilodeRoock]

43:03 - No offense meant, but lecturing from a book in hand?? - There are universities where the students would insist that the instructor / professor would 110% OWN (mentally) the books and topics to be studied in class.

[premkumar-soff]

I like many of your course playlists but like to know is there a topology course from your list which serves slightly advanced than this introduction which covers examples from algebra analysis in topology course

[StarsManny]

I'm not religious but I thought the prayer was really nice and it gave a good atmosphere!

[allyourcode]

@46:56 I feel like this fact deserves way more emphasis. To me, this fact really "justifies" topology as an area of study in its own right, separate from metric spaces, which are more concrete and easier to visualize.

[waverly]

I started watching some topology lecture vids a year ago. The subject matter was more relevant than I expected, when I was studying complex analysis and they were talking about areas around a point in Cauchy's integral theorem.

[darioniero]

Hi, I'm an italian physic's student. I've studied Calculus and Linear algebra. what are the prerequisites for this course?

[NilodeRoock]

09:25 - Why not handle this issue forever by mentioning Peano's axioms. Or explain that when the integers are constructed from the natural numbers it becomes "painfully" clear that 0 can't be a natural number.

[ABC-jqve]

This lecture is so great, thank you!!

[jonathantraugott]

Line two of the distance function should say d(x, y) = 0 iff x = y. d(x, y)>=0 and d(x, x) just gives a pseudometric.