Identifying Open, Closed, and Compact Sets | Real Analysis Exercises

Показать описание

We look at six sets and determine if each one is either open, closed, compact, or any combination of the three. We will use multiple definitions in our solutions. An open set is a set whose elements all have some neighborhood around them completely contained in the set. A closed set is a complement of an open set, or equivalently a set whose limit points are all contained in the set. A compact set is closed and bounded, or equivalently a set whose open covers all have finite subcovers. #realanalysis

Thanks Summatin for the chapters!
0:00 Intro
0:20 Integers
2:07 Union of reciprocals of natural numbers with set containing zero
3:20 Real numbers
4:05 Union of open and closed intervals
5:01 Rational numbers
5:42 Singleton set containing only seven
6:23 Real numbers revisited

◉Textbooks I Like◉

★DONATE★

Thanks to Petar, dric, Rolf Waefler, Robert Rennie, Barbara Sharrock, Joshua Gray, Karl Kristiansen, Katy, Mohamad Nossier, and Shadow Master for their generous support on Patreon!

Follow Wrath of Math on...

Рекомендации по теме

Комментарии

Love the channel! One quick correction: 7 is not a limit point of {7}, at least according to the definition given in Understanding Analysis.

curiouslockpicker

I'm studying point set topology and hadn't been taught compactness in real analysis before (since it's covered in topology as an optional module) and I really struggled getting to grasps with it. its nice to see you go through plenty of examples in the euclidean topology that can act as a referral point. So thank you!

samuelhawksworth

nailed 5, maybe more of these? These kinds of examples are really indispensable. Thanks a ton

soumyadipghosh

0:20: Integers
2:07: Union of reciprocals of natural numbers with set containing zero
3:20: Real numbers
4:05: Union of open and closed intervals
5:01: Rational numbers
5:42: Singleton set containing only seven
6:23: Real numbers (again)

summatim

Nice examples and discussions of reasons.

TranquilSeaOfMath

Hey, love the examples!! Just wanted to ask where is the video that explains that a set is compact if it is closed and bounded?

okikiolaotitoloju

Identifying Open, Closed, and Compact Sets | Real Analysis Exercises

Identifying Open, Closed, and Compact Sets | Real Analysis Exercises

Real Analysis 13 | Open, Closed and Compact Sets

Open Covers, Finite Subcovers, and Compact Sets | Real Analysis

Understanding Compact Sets

Show that (0, 1] is not compact - Topology - Compact sets

Real Analysis 13 | Open, Closed and Compact Sets [dark version]

Open Set, Closed Set, Compact Set, Dense Set, Perfect Set, Connected Set: Real Analysis| Topology-8

Compact Sets Contain Supremum and Infimum | Real Analysis Exercises

An Introduction to Compact Sets

Functional Analysis 16 | Compact Sets

Lecture 2B : Open sets, closed sets and compact sets

Compact Sets and Open Covers, Real Analysis II

Real Analysis ep19: Compact Sets (Oct 21, 2022)

Lecture 3: Compact Sets in Rⁿ

Demo Configurator - Compact drop-down and lift-up Opening System 1.0

Compact iff Sequentially Compact full proof, Real Analysis II

Real Analysis: Compact Set of Real Numbers

402.3X1 Local Global Heartbreak: A Motivation for Compact Sets

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

Closed subset of compact set is compact - Lec 61 - Real Analysis

Is Intersection of two compact set is compact

402.3X5 Compact, Closed, and Bounded: The Heine-Borel Theorem

Real analysis.#Which set is Compact ?? Identify from the given figure.# CSIR UGC NET, GATE, IIT-JAM

Limit Points and Compact sets syllabus updates #Shorts