[old series] General Topology Lecture 16 Part 1

Sixteenth lecture in general topology. Topics include sigma algebras, measureable spaces, Borel sets and the Borel sigma algebra, measure spaces, Borel measure, and Lebesgue measure. This lecture is in five parts.

[RecursiveRecursive]

In any future comments, I will seek to use the terminology defined in this series. This series is very self contained. In fact, this is currently the best series of online topology lectures available. I am grateful for the time and effort you have put into these courses. I hope to see more in the future. Thanks.

[jajipur]

Thank you for clarifying this.
Best,
Jajipur

[RecursiveRecursive]

At 21:00 it is said:
"Let V open in the plane. Then V = A x B where A, B are subsets of the real line."
But consider V = { (x, y) : y < x*x }.
Now V is open in the plane. But V can not be written as a cross product of open sets in the real line.
Have I misunderstood something?

[RecursiveRecursive]

Thank you. I now see why each open set in the plane is a union of basis elements. However, it still remains to be shown that each open set can be written as a *countable* union of basis elements. Is this needed to show measurability? In regards to the n-dimensional interval, see Rudin (3rd ed.) definition 11.4.

[wendywang]

at 20:13, the function f:R->R^2 should be f: X->R^2.

[RecursiveRecursive]

Is a "basic open set" the same thing an "open interval" in the plane? Do you have a proof that "each open set in the real plane is either a basic open set or a union of basic open sets?" From what I understand, it involves both the countability of the rationals and the density of the rationals in the reals.

[logician]

So then, Topology would be necessary for scientists who are trying to develop a unified field theory of physics, since geometry holds sway over all the universal forces???

[jajipur]

Hi, I am a novice here. But could you explain a bit more on what you mean by a mble functions in X when you state your third theorem in this posting? As per your definition there should be a topological space Y such that f maps X to Y. I understand that Y may be arbitrary. But is it standard to assume that such a Y exists when you state f is mble on X. I am sorry if I am being really silly here. But thank you for such wonderful lessons.

[logician]

Uhh ok, so topology would be an important part of solving the Riemann Hypothosis???

[logician]

I wish I knew what to use all of this for...LOL

[johncocksmith]

Did you say that Sigma comes from the French word Summe???
You realize that Sigma comes from Ancient Greece right? and probably the Phoenicians before that.