We introduced closed sets and clopen sets. We'll visit two definitions of closed sets. First, a set is closed if it is the complement of some open set, and second, a set is closed if it contains all of its limit points. We see examples of sets both closed and open (called "clopen sets") and we see examples of sets that are neither closed nor open. We complete several quick proofs that a set is closed. We see why the empty set is closed, and why the real numbers are closed. #realanalysis
Open Sets: [ Ссылка ]
Proof Union of Open Sets is Open: [ Ссылка ]
Limit Points: [ Ссылка ]
Proof for Equivalent Definitions of Closed Sets: [ Ссылка ]
Real Analysis Course: [ Ссылка ]
Real Analysis exercises: [ Ссылка ]
Table of Contents
00:00 Intro
00:29 Defining Closed Sets with Open Sets
01:34 Closed Set Examples
04:40 Defining Closed Sets with Limit Points
05:59 More Closed Set Examples
06:48 A Set Neither Closed Nor Open
08:58 Closure of a Set
09:00 Showing Closed Intervals are Closed Sets Using Limit Point Definition
◉Textbooks I Like◉
Graph Theory: [ Ссылка ]
Real Analysis: [ Ссылка ]
Proofs and Set Theory: [ Ссылка ] (available for free online)
Statistics: [ Ссылка ]
Abstract Algebra: [ Ссылка ]
Discrete Math: [ Ссылка ]
Number Theory: [ Ссылка ]
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