18.100B Real Analysis (Spring 2025, MIT OCW): Lecture 13 - Open and Closed Sets; Coverings; Compactness

18.100B Real Analysis

18.100B Real Analysis (Spring 2025, MIT OCW). Instructor: Prof. Tobias Holck Colding. This course gives an introduction to analysis, and the goal is twofold:

1. To learn how to prove mathematical theorems in analysis and how to write proofs.

2. To prove theorems in calculus in a rigorous way.

The course will start with real numbers, limits, convergence, series and continuity. We will continue on with metric spaces, differentiation and Riemann integrals. After that, we will move on to differential equations. (from ocw.mit.edu)

Lecture 13 - Open and Closed Sets; Coverings; Compactness

We begin by giving a useful characterization of what it means for a set to be closed. This is the notion that a set contains all its limit points. After that, we turn to the notion of a cover and, in particular, open covers and use it to define compact subsets of a metric space.

Go to the Course Home or watch other lectures:

Lecture 01 - Introduction to Real Numbers

Lecture 02 - Introduction to Real Numbers (cont.)

Lecture 03 - How to Write a Proof; Archimedean Property

Lecture 04 - Sequences; Convergence

Lecture 05 - Monotone Convergence Theorem

Lecture 06 - Cauchy Convergence Theorem

Lecture 07 - Bolzano-Weierstrass Theorem; Cauchy Sequences; Series

Lecture 08 - Convergence Test for Series; Power Series

Lecture 09 - Limsup and Liminf; Power Series; Continuous Functions; Exponential Function

Lecture 10 - Continuous Functions; Exponential Function (cont.)

Lecture 11 - Extreme and Intermediate Value Theorem; Metric Spaces