All About Closed Sets and Closures of Sets (and Clopen Sets) | Real Analysis

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:    • Intro to Open Sets (with Examples) | ...  
Proof Union of Open Sets is Open:    • Proof for Unions and Intersections of...  
Limit Points:    • Limit Points (Sequence and Neighborho...  
Proof for Equivalent Definitions of Closed Sets:    • A Set is Closed iff it Contains Limit...  

Real Analysis Course:    • Real Analysis  
Real Analysis exercises:    • Real Analysis Exercises  

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

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