Question

Prove that a set $M$ is closed if and only if $\bar{M}=M$.

   Prove that a set $M$ is closed if and only if $\bar{M}=M$.

Single Variable Differential and Integral Calculus: Mathematical Analysis

Single Variable Differential and Integral Calculus: Mathematical Analysis

Elimhan Mahmudov 1st Edition

Chapter 1, Problem 25

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A set $M$ is closed if it contains all its limit points. The closure of a set $M$, denoted by $\bar{M}$, is the union of $M$ and the set of all limit points of $M$. Show more…

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Prove that a set $M$ is closed if and only if $\bar{M}=M$.

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Key Concepts

Closed Sets

A closed set in topology is one that contains all its limit points. This means that if a sequence (or more generally, a net) of points within the set converges to a point, that limit point must also be part of the set. This property makes closed sets 'complete' in the sense that they include all accumulation behavior within themselves.

Closure of a Set

The closure of a set is defined as the smallest closed set that contains the original set. It is obtained by adding all the limit points of the set to the set itself. This concept is fundamental because it provides a systematic way of 'completing' a set with respect to limit points, bridging the gap between arbitrary sets and closed sets in a topological space.

Limit Points

A limit point of a set is a point such that every neighborhood around it contains at least one point from the set other than itself. Recognizing limit points is central to understanding closures and closed sets, as the closure of a set is precisely the union of the set and all of its limit points.

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