Intro Real Analysis, Lec 31: Open Sets on the Real Line, Continuity & Preimages of Open Intervals

Показать описание

Introduction to Real Analysis Lecture 31. Topology Unit, Part 1.

(0:00) Comments about content and the final exam.
(2:01) Today will be topology of the real line. The last two weeks we will use a more abstract approach involving metric spaces as well and apply the ideas to studying recursively defined sequences.
(3:13) An important property of open intervals (given any point in an open interval, there is an open interval centered at the point that is entirely contained in the given open interval) and give proof.
(14:07) Extend the property to unions of open intervals and give proof.
(22:47) Extend the property to the intersection of a finite number of open intervals and give proof.
(33:40) Definition of an open set (in the set of real numbers).
(36:13) Mention that any nonempty open set (in R) is the union of open intervals. Any such set can even be proved to be the union of a countable collection of disjoint open intervals.
(39:18) Theorem: Given a real-valued function defined on R, the function is continuous on R if and only if the preimage of any open interval is an open set.
(44:40) Sketch of proof.

AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.

Рекомендации по теме

Комментарии

I am studying for a master's degree in applied mathematics and I had a lot of trouble understanding these topics. Thank you very much for your video. I have finally understood this topic. You are very good at explaining. Greetings from Mexico.

nemachtianimx

Thanks so much for making these lectures publicly available. I'm passing my Intro to Real Analysis thanks to your help!! :)

richardvazquez

Applied Math Major here, and also has MS in Stats. I would love to sit in your class. I love how you teach Topology with drawings.

yaweli

Thank you very much! I am learning Real analysis and find open set hard to understand. Your video is very helpful!

bowlofsoba

Cheers from South Africa, I’m writing final year exam for real analysis. Uniform continuity would be the next lecture I’d like to see Prof. Great upload🔥

damianboone

Thank you so much sir. Your explanation is very lucid.

abhishekpaul

At 9:07, isnt r <= b-x and r <= x-a since its the minimum. Example r= min(2, 3). so r = 2. so r <= 2 and r <= 3.

algorithmo

I just wanted to say your lectures are unbelievably helpful! It has really been how I've understood my course. Out of curiosity, what book are you teaching from?

lanefarrel

Which year of university is this lecture taught

algorithmo

Sir, is there any chance you can upload linear algebra and topology lectures?

ritesharora

Intro Real Analysis, Lec 31: Open Sets on the Real Line, Continuity & Preimages of Open Intervals

Intro Real Analysis, Lec 31: Open Sets on the Real Line, Continuity & Preimages of Open Interval...

Intro Real Analysis, Lec 29: The Most Beautiful Equation in the World, Taylor Series Calculations

Intro Real Analysis, Lec 32: Open and Closed Sets in the Real Line and in the Plane

Several examples on neighborhood on reals - Lec 31 - Real Analysis

Intro Real Analysis, Lec 37, Part 1: Taylor Series, Cantor Sets & Logistic Map, Review Topology

Real analysis(lec 31)(sem2) (realnumberchapter 1)epsilon neighborhood e.g#sachinchandra#(9693967379)

Lec-31 M Test for Uniform Convergence of sequence and series of functions. Real Analysis II. Rudin.

Intro Real Analysis, Lec 35: Sup Norm and Metric on C[a,b], Sequence Space, Open & Closed Sets

Intro Real Analysis, Lec 33: Euclidean Metric, Triangle Inequality, Metric Spaces, Compact Sets

Intro Real Analysis, Lec 17: Mean Value Theorem Corollaries, Definition of Riemann Integral

Intro Real Analysis, Lec 37, Part 2: Review Topology, Quick Review of Other Real Analysis Topics

Intro Real Analysis, Lec 19, Part 3: Review for Real Analysis Exam 2

Intro Real Analysis, Lec 36: Chaos, Logistic Map, Period Doubling, Symbol Space, Shift Map

Intro Complex Analysis, Lec 31, Laurent Series, Poles of Complex Functions, Essential Singularities

Intro Real Analysis, (Most of) Lec 14: Uniform Continuity Non-Examples, Variation of a Function

Intro Real Analysis, Lec 28: Taylor Series for sin(x), cos(x), e^(x), Uniform Convergence

Intro to Real Analysis, Lec 4: Cardinality, Cantor, Continuum Hypothesis, Ping Pong Ball Conundrum

Intro Real Analysis, Lec 9: Recursive Sequences, Limit Superior & Inferior Definitions & Pro...

cantor's ternary set| cantor set| countability of sets| real analysis | lec#06

Real Analysis I (lec. 3.3.D) Uniform Continuity : Open, Closed, Compact Sets

Real Analysis I (lec. 4.2.C) The Algebra of Derivatives

Real Analysis I (lec. 1.2.C) Cauchy Sequences

Lec 31 : Adaptive Filters 3 : Convergence of the LMS algorithm

Real Analysis I (lec. 0.2.A). Relations and Functions.