I understand there is some concerns with the definitions I used toward the end of the video; particularly about neighborhoods and limit points. Depending on what textbook you use, and how in-depth the author wants to get, these definitions will vary, which is understandably annoying. The definitions I used were taken from Kolmogorov and Fomin's book on Real Analysis which develops topology just enough to discuss functional analysis and measure theory. It is not a dedicated textbook in topology and is basically all the topology I know. Hopefully in the fall I will get more in-depth about the wonderful world of topological spaces.
Can you elaborate on this, how did your experiences dealing with schizophrenia affect how you learned math? what type of math were you doing? I've heard of some people having very unique, but insightful, perspectives in academia from schizophrenia. I'm not trying to glorify your schizophrenia, I'm just curious as to your experience.
I'm a computer scientist but I'm getting a deeper understanding of the fundamental mathematical structures to further refine my thought process. This channel is really an ice breaker for a lot of unknown topics, or topics that I don't fully have a grasp on just yet. As I work my way to a PhD myself I'm seeing more and more abstractions that start leaking into pure maths and I found a ton of stuff that I started studying myself so it makes sense to go the extra mile if you're passionate keep up the good work
Topology was really cool. It was really cool to see topics like continuity and open sets completely fleshed out in a general manner without using metrics and whatnot. Also, I am excited for your measure theory series!
Munkres is not a good book when initially reading topogy. "Topology without tears" is the book one should start with. Best topology book I have ever read.
I reckon that Functional Analysis has a strong connection to Topology, and Fourier analysis in particular involves Topological groups, so i had assumed you had ur hands in it for while. But then again the lvl of Topology involved isn't that thorough, at least anything paper 1 covered is more than adequate.
Im currently doing my p4 edexcel IAL course ( year13 ) . Love your vids , theyre so interesting although im struggle to understand most of the topic 😂❤
I am using that exact book measure theory, measure and fractal analysis for my undergrad capstone project! I am having a hard time with it, and worried sick I won’t accomplish anything :( interesting video!
I just finished up my undergrad final year project and I used this book as well! “Fractals Everywhere” by Michael Barnsley makes some of the concepts more digestible, depending on what you’re focusing on. Best of luck!
@@marktobin3638 oh wow I really appreciate that, I’m mainly just looking into defining fractals recursively. I haven’t taken topology I took differential geometry instead, so the measure and topological aspects are new to me and overwhelming to say the least. I’ll look into the recommendation I appreciate it!
Read "Topology without Tears" first. Best book on topology i have ever read. After that you can read the other books such as the one written by munkres.
I find it interesting that in America you only take topology as a graduate course, whereas here in the UK we had it as a second year undergraduate option. This stuff isn't just abstract nonsense either--you need it if you want to study things like the fundamental groups of topological spaces with covering spaces and whatnot.
Let me guess...it is because the UK is far superior in science, technology and mathematics (maths...so lame!) than the US in every way. Way to go Brits!!
@@Corredephd Usually there is no calculus course in european universities. Straight into real analysis. I have taken the contents of this video and much more in first year undergraduate analysis.
Yes you need topology to study fundamental groups of topological spaces lol. Topology is an undergraduate course just about everywhere to my knowledge. He didn't take it because he didn't finish his math undergraduate.
Being in highschool I know nothing about mathematics at your level, I would like to ask whether math is so disconnected up there that its better to say for some mathematicians, "I'm an algebraic geometer" rather than "I'm a mathematician" while talking to, say, physicists.
You definitely specialize at this level. Within pure math, there is algebraists and analysts (what I do.) But it breaks down more than that, since algebra is made up of group theory, ring theory, field theory, etc. So mathematician isn't very descriptive but it is good enough when speaking with people outside of mathematics.
For me, topology feels more “concrete” than algebra. I like seeing and using algebraic structures that arise from other mathematical structures. But, pure algebra itself seems more abstract than topology. I would place algebra to the left of topology on your pure-applied spectrum. But, that’s just me. There are whole areas of math such as geometric group theory where this situation is turned upside down.
I love topology. It’s been my favorite subject since the first time I took an undergraduate topology course. Somehow, the way of thinking comes naturally to me. I have often heard the opinion of some mathematicians that the topology used directly in analysis is the only topology anyone needs to know. I disagree, because there is a whole world of topology outside of analysis, and it can be applied to many situations in other areas of pure and applied math. The more common definition of a limit point of A is any point x for which every neighborhood of x contains at least one element of A that is not x. Your definition says that every neighborhood of x must contain infinitely many elements of A for x to be considered a limit point. I somewhat dislike the definition you are using, because it doesn’t work for topologies in which some pairs of points share all the same neighborhoods. But, I can’t tell people how to do topology.
Topology is actually a very aplicable field, it has tons of connections with combinatorics, geometry, cs. The thing is it's foundation is very abstract and general, so at first it might seem very strange but it actually has strong relationships with tons (and progressively more) fields. I think the issue is that at first you focus on 'bad' topological spaces because you build it from the bottom up but in reality you tend to work mostly with very nice topological spaces (manifolds, wedges of nice spaces, finite topological spaces, simpliical complexes, cw complexes, and a long etc.) that arise naturally from studying relatively concrete problems. Don't get me wrong, the whole categorical angle of the discipline is inevitable, so better get used to talking about "the cylinder of a funcion" and other fun nonsense. : ) Also feel obligated to say that topology and set theory are very very different. Set theory is deeply terrifying in comparison.
Your definition of neighborhood is actually that of an open neighborhood. For it to be a general neighborhood, it need not be in the topology, but it must contain an open set O that itself contains the point x. I.e., U_x is a neighborhood of x [(1) x∈U_x AND (2) ∃O∈τ s.t. x∈O⊆U_x]. Also, your definition of a limit point is a special type of limit point, called an ω-limit point. Generally, x is a limit point of a set A iff every neighborhood of x contains at least one point of A different from x itself. In other words, x is a limit point of A ∀U_x neighborhood of x, ∃y∈( U_x ∩ (A\{x}) ).
Some authors require neighborhoods to be open, although, yes, it is more common to just require that the neighborhood contains an open set containing the point. It's also important to note that there's a bunch of words (limit point, cluster point, accumulation point, etc.) that all mean something along the lines of what you call an omega-limit point. Which word means what or even if they're synonymous depends highly on the author. It's actually super annoying when you open a textbook, see the word limit point, and have to go search the book to see what the author considers it to be.
@@andrewhaar2815True. But what I have written down are, from my experience, the most prevalent and widely-used terms. Hence, my comment. I also don't like the fact that the words are not standardized and every author just uses what he likes. Regarding the neighborhoods - yes, some do require them to be open. Regarding the limit point - from what I've seen, real analysis books whose authors define limit point of A as having the property that every neighborhood of it contain infinitely many members when intersected with A, are often more applied or undergraduate-leaning, whereas higher-level books (that go a bit further into pure topology) use the definition I stated.
This is all definitions. Limit points and neighborhoods are defined differently in almost every book I have ever read. I don’t think there’s really any right or wrongs.
The notion of “pure” is inherently ambiguous in mathematics. Pure sugar has a clear meaning. Perhaps by pure you mean something like if a word like pendulum or inductor appears in the problem it becomes applied versus pure.
I understand there is some concerns with the definitions I used toward the end of the video; particularly about neighborhoods and limit points. Depending on what textbook you use, and how in-depth the author wants to get, these definitions will vary, which is understandably annoying. The definitions I used were taken from Kolmogorov and Fomin's book on Real Analysis which develops topology just enough to discuss functional analysis and measure theory. It is not a dedicated textbook in topology and is basically all the topology I know. Hopefully in the fall I will get more in-depth about the wonderful world of topological spaces.
The fact that I can more or less understand what you're saying says a lot about how schizophrenia overtook my brain during math classes💀
Can you elaborate on this, how did your experiences dealing with schizophrenia affect how you learned math? what type of math were you doing?
I've heard of some people having very unique, but insightful, perspectives in academia from schizophrenia. I'm not trying to glorify your schizophrenia, I'm just curious as to your experience.
@@davidcarter8269 It was supposed to be a joke about how learning maths makes you go insane
@@lasterbitz4490 If you choose to specialize math in Uni you're already a part of a small minority of masochists lol
@@okplay9446 Ong lmao
I'm a computer scientist but I'm getting a deeper understanding of the fundamental mathematical structures to further refine my thought process. This channel is really an ice breaker for a lot of unknown topics, or topics that I don't fully have a grasp on just yet. As I work my way to a PhD myself I'm seeing more and more abstractions that start leaking into pure maths and I found a ton of stuff that I started studying myself so it makes sense to go the extra mile if you're passionate
keep up the good work
couldn't relate more
Topology was really cool. It was really cool to see topics like continuity and open sets completely fleshed out in a general manner without using metrics and whatnot. Also, I am excited for your measure theory series!
Most mathematician Don't even understand topology 🤔 very weird
I'm currently taking a topology class as 4th semester undergrad I'm totaly loving it!
Man, point set topology, together with category theory are the most terse subject in math that I have encountered.They always make my head spin...
You should try Topology by Munkres, if your course is not going to be using it already! It’s really great, probably the best book I have had!
I will check it out :)
I personally think Munkres is too much to learn from. But maybe that's because I do algebra so I use very little topology.
Munkres is not a good book when initially reading topogy. "Topology without tears" is the book one should start with. Best topology book I have ever read.
@@algebraist3212 What do you mean? Topology appears in number theory and algebraic/arithemtic geometry.
I reckon that Functional Analysis has a strong connection to Topology, and Fourier analysis in particular involves Topological groups, so i had assumed you had ur hands in it for while. But then again the lvl of Topology involved isn't that thorough, at least anything paper 1 covered is more than adequate.
Im currently doing my p4 edexcel IAL course ( year13 ) . Love your vids , theyre so interesting although im struggle to understand most of the topic 😂❤
I was wondering about the similarities and differences between the definitions of topological spaces and sigma algebras. That's really interesting!
I am using that exact book measure theory, measure and fractal analysis for my undergrad capstone project! I am having a hard time with it, and worried sick I won’t accomplish anything :( interesting video!
I just finished up my undergrad final year project and I used this book as well! “Fractals Everywhere” by Michael Barnsley makes some of the concepts more digestible, depending on what you’re focusing on. Best of luck!
@@marktobin3638 oh wow I really appreciate that, I’m mainly just looking into defining fractals recursively. I haven’t taken topology I took differential geometry instead, so the measure and topological aspects are new to me and overwhelming to say the least. I’ll look into the recommendation I appreciate it!
Read "Topology without Tears" first. Best book on topology i have ever read. After that you can read the other books such as the one written by munkres.
Would be interesting to hear what you think of topos theory
16:14
i think this topology is called the discrete topology and this is the trivial one {X,Φ}
One is called the discrete topology and the other one is called indiscrete topology but both are just trivial.
I find it interesting that in America you only take topology as a graduate course, whereas here in the UK we had it as a second year undergraduate option. This stuff isn't just abstract nonsense either--you need it if you want to study things like the fundamental groups of topological spaces with covering spaces and whatnot.
Let me guess...it is because the UK is far superior in science, technology and mathematics (maths...so lame!) than the US in every way. Way to go Brits!!
There's an undergraduate and graduate level course of topology offered at most universities in America
I guarantee you that the distance between an undergraduate topology course and a graduate one is as much as calculus is distant from real analysis
@@Corredephd Usually there is no calculus course in european universities. Straight into real analysis. I have taken the contents of this video and much more in first year undergraduate analysis.
Yes you need topology to study fundamental groups of topological spaces lol. Topology is an undergraduate course just about everywhere to my knowledge. He didn't take it because he didn't finish his math undergraduate.
11:05 there are also clopen sets
Being in highschool I know nothing about mathematics at your level, I would like to ask whether math is so disconnected up there that its better to say for some mathematicians, "I'm an algebraic geometer" rather than "I'm a mathematician" while talking to, say, physicists.
You definitely specialize at this level. Within pure math, there is algebraists and analysts (what I do.) But it breaks down more than that, since algebra is made up of group theory, ring theory, field theory, etc. So mathematician isn't very descriptive but it is good enough when speaking with people outside of mathematics.
For me, topology feels more “concrete” than algebra. I like seeing and using algebraic structures that arise from other mathematical structures. But, pure algebra itself seems more abstract than topology. I would place algebra to the left of topology on your pure-applied spectrum. But, that’s just me. There are whole areas of math such as geometric group theory where this situation is turned upside down.
I love topology. It’s been my favorite subject since the first time I took an undergraduate topology course. Somehow, the way of thinking comes naturally to me.
I have often heard the opinion of some mathematicians that the topology used directly in analysis is the only topology anyone needs to know. I disagree, because there is a whole world of topology outside of analysis, and it can be applied to many situations in other areas of pure and applied math.
The more common definition of a limit point of A is any point x for which every neighborhood of x contains at least one element of A that is not x. Your definition says that every neighborhood of x must contain infinitely many elements of A for x to be considered a limit point. I somewhat dislike the definition you are using, because it doesn’t work for topologies in which some pairs of points share all the same neighborhoods. But, I can’t tell people how to do topology.
Topology is actually a very aplicable field, it has tons of connections with combinatorics, geometry, cs. The thing is it's foundation is very abstract and general, so at first it might seem very strange but it actually has strong relationships with tons (and progressively more) fields. I think the issue is that at first you focus on 'bad' topological spaces because you build it from the bottom up but in reality you tend to work mostly with very nice topological spaces (manifolds, wedges of nice spaces, finite topological spaces, simpliical complexes, cw complexes, and a long etc.) that arise naturally from studying relatively concrete problems. Don't get me wrong, the whole categorical angle of the discipline is inevitable, so better get used to talking about "the cylinder of a funcion" and other fun nonsense. : )
Also feel obligated to say that topology and set theory are very very different. Set theory is deeply terrifying in comparison.
Your definition of neighborhood is actually that of an open neighborhood. For it to be a general neighborhood, it need not be in the topology, but it must contain an open set O that itself contains the point x.
I.e., U_x is a neighborhood of x [(1) x∈U_x AND (2) ∃O∈τ s.t. x∈O⊆U_x].
Also, your definition of a limit point is a special type of limit point, called an ω-limit point. Generally, x is a limit point of a set A iff every neighborhood of x contains at least one point of A different from x itself. In other words, x is a limit point of A ∀U_x neighborhood of x, ∃y∈( U_x ∩ (A\{x}) ).
Some authors require neighborhoods to be open, although, yes, it is more common to just require that the neighborhood contains an open set containing the point.
It's also important to note that there's a bunch of words (limit point, cluster point, accumulation point, etc.) that all mean something along the lines of what you call an omega-limit point. Which word means what or even if they're synonymous depends highly on the author. It's actually super annoying when you open a textbook, see the word limit point, and have to go search the book to see what the author considers it to be.
@@andrewhaar2815True. But what I have written down are, from my experience, the most prevalent and widely-used terms. Hence, my comment. I also don't like the fact that the words are not standardized and every author just uses what he likes.
Regarding the neighborhoods - yes, some do require them to be open.
Regarding the limit point - from what I've seen, real analysis books whose authors define limit point of A as having the property that every neighborhood of it contain infinitely many members when intersected with A, are often more applied or undergraduate-leaning, whereas higher-level books (that go a bit further into pure topology) use the definition I stated.
This is all definitions. Limit points and neighborhoods are defined differently in almost every book I have ever read. I don’t think there’s really any right or wrongs.
It's also noteworthy to mention that if the space is Hausdorff that definition works fine
It’s OK to define a neighborhood as being open. It’s a simpler definition. I prefer it.
I struggled in topology and got a B-.
I found Group Theory the purest form - but that is just me.
yo, bro, i am currently studying Fields and Galois Theory this semester, how was that for you lol
Would you say a computer science degree is easier than pure mathematics?
From which university are you doing your PhD
Kent State
The notion of “pure” is inherently ambiguous in mathematics.
Pure sugar has a clear meaning.
Perhaps by pure you mean something like if a word like pendulum or inductor appears in the problem it becomes applied versus pure.
You should get some sleep its 2 am.. my pc died ;c