Introduction to Higher Mathematics – Lecture 14: Topology
[youtube http://www.youtube.com/watch?v=FHL4udeLf9Q&fs=1&rel=0]
Here we delve a bit into the realm of topology, adding a notion of “closeness” to our sets and seeing how that helps us understand the real numbers.
[youtube http://www.youtube.com/watch?v=6v0hk4FGTDU&fs=1&rel=0]
Its all mathematics, you know the deal.
Hi Bill, thank you for making this series, I really enjoy it. I caught a
little typo at around 15:25, where the slide shows: “The union of an
infinite number of closed sets may not be OPEN!” – but you mean to say
CLOSED.
Maybe you caught this already, I don’t mean to nitpick 🙂
Again, thank you for taking the time to make this series!
11:45
“The intersection of an infinite number of open sets may not be open.”
As n –> infinity, the (-1/n, 1/n) approaches {empty}, which makes the
intersection closed since || epsilon || >0. So the core idea here is (0,0)
is {empty}.
15:22
“The union of an infinite number of closed sets may not be closed.”
the [1/n-1, 1-1/n ] will never equal to [1, 1] as n -> infinity, which
makes the union open, (-1,1).
Correct me if i’m wrong, thanks.
So.. if I got it right, Q is disconnected because it is essentially not the
full line of numbers.
Is it totally disconnected? I kept hearing that between each and every
fraction there is an infinite amount of transcendental numbers (that is
what you call RQ right?)
Bill fantastic series! Are your slides online in PDF or PPT format?
Hello Bill,
Did you get the chance to talk on Prime numbers?
I think it is very handy for Computer science.
Sham
Is the cluster point synonymous with a limit point Bill?
Early happy Birthday Mr.Shillito. Thanks for the lectures. I can’t for
Abstract Algebra!
My mistake. I must have written down the wrong thing, thank you for
catching that – the narration is correct, I’ll fix the visual with an
annotation. This particular PPT took me a good bit longer to make, so I
hope I didn’t inadvertently make any more errors. Thank you for catching
this!
though, probably I do not commented in my last paragraph, this is the
introduction I was expecting to start my mathematical education again
because; math are the love of my life. Thanks Shillito and excellent
intervention in all the program. Keep in mind infinity – one is not and
cannot be infinity, Thanks and keep up the good works.
Thank you for these Professor Shillito. I always get so excited when you
put out a new video =D
at 29:00 you narrate “… and the intersection of U and S union the
intersection of V and S is the entire set S”, however on the slide we see U
∪ V = S, but not (U ∩ S) ∪ (V ∩ S) = S
Thank you! Abstract Algebra starts on Lecture 16 … stay tuned! 😉
It seems the story with the Cantor set is the same as with the infinite
ternary tree. There’s a countable number of nodes (segments here) and
levels but the paths (the resulting points here) can’t form a countable
set. Each path, or each point in the Cantor set, can only be constructed by
an inifinite sequence of specifications. That’s problematic for non
Platonists. It’s not clear that the paths (or the points in the Cantor set)
are all given once the tree (or Cantor’s procedure) is given.
Did Euclid have a different point of view? He says, “The ends of a line are
points”, so apparently he thinks a line is composed of points. He also
says, “Things which coincide are equal”. But all that matters to him, as
far as a line segment, is length, or Lebesgue measure. So it would seem
that [0,1] and [0,1) are the same thing! What is interesting about [0,1) is
that the boundary point is OUTSIDE the set; i.e. the set is only defined in
terms of a ‘universe’ of points outside the set.
Thanks a lot.. limit/cluster point part was especially helpful, I was not
getting the concept right earlier.
Once again, hats off for another clear and concise lecture!
I really have a hard time understanding how the cantor set is
uncountable… My first thought is that there should be no problem to just
count them row by row, like 0,1,1/3,2/3,1/9,2/9,7/9,8/9, and so on. I don’t
see how this is different from the method of counting the rational numbers,
which brings me to my second thought: How can a subset of the rational
numbers be uncountable? I don’t see what I’m getting wrong, could you
please try to explain it further to me?
What you’re actually counting is not the Cantor set itself, but actually
just the endpoints of the intervals that are left over after those
deletions! For instance, after the first interval is deleted, your
endpoints are {0, 1/3, 2/3, 1}, indeed a countable (finite) subset of the
rational numbers, but the closed intervals [0, 1/3] and [2/3, 1] still have
uncountably many points.
I love this song. Listened to it dozens of time. But I gotta say, I don’t
like the condescending way he says “and even the gringos” in the very
beginning. Does this bother anyone else?
It’s simple mathematics.
Why Did One Straw Break The Camel’s Back?
it IS all mathematics. Mathematics is the solution for most problems.
This album is so amazin’
1+1=3 for sufficiently large values of 1
15 years later…. and mos def predicted the future
this is some lyrical high stand
mind fuck just think about every word
OMFG
i really don’t understand the ”Bucka bucka bucka bucka bucka”.
+NuttyNuttinMusic LOL. Riiight?!
damn hella lyrical
https://soundcloud.com/ill_will_hs/jedi-mathematics-boosted
Lyrics: rapgenius.com/Will-hs-jedi-mathematics-lyrics
one of the illest tunes out there
First time where I got goosebumps just by reading the lyrics, especially
the second verse.
Mos Def knows how to rap. Like damn 
i’m going learn mathematics! 🙂 (y)
is dj premier laying the beats on this track?
this is dope 
“Crack Mothers, crack babies and AIDS pateints, youngbloods can’t spell but
they can rock you in Playstation…”
14 YEARS LATER
Gang Members, gang leaders and caps popped, youngbloods (STILL) can’t add
but the could rock you playin’ Xbox …
Am I wrong?
Mos Def – “Mathematics”
Why did one straw break the camel’s back?
Here’s the secret:
The million other straws underneath it.
It’s all mathematics.
Madden 02, who remember this shit, oh shit that brings back memories!