# Comparing PERFECT SET vs COMPLETE SET | Verifying Compact Sets

• Ka Yan
In summary, a "perfect set" in topology is a non-empty, closed set with no isolated points, while a "complete set" may have different definitions depending on the field of mathematics. To verify if a set is compact, one method is to show that every infinite sequence in the set has a subsequence that converges to a point in the set. Another way is to check if every open cover of the set has a finite subcover. This definition of a compact set is general, but there is also a specific definition for subsets of Euclidean spaces.
Ka Yan
1. What's the difference between PERFECT SET and COMPLETE SET? Can I have an explicit explain to it, rather than a discribtion from definition?

2. How can I verify whether a set is compact or not more evidently and effective?

Tks!

Last edited:
Ka Yan said:
1. What's the difference between PERFECT SET and COMPLETE SET? Can I have an explicit explain to it, rather than a discribtion from definition?
First, are you talking about topology? From your second question, I guess that you are but there are notions of 'perfect' and 'complete' sets for many different fields of mathematics. Topologically, a set is "perfect" if it is non-empty, closed, and has no isolated points. I don't recognize "complete set" in topology and cannot find any reference to it. Plenty of references to "complete set" of different kinds of things!
Do you have a definition of "complete set"?

2. How can I verify whether a set is compact or not more evidently and effective?

Tks!
Precisely what is your definition of "compact" set? One way to determine if a set is compact is by showing that every infinite sequence of points in the set has a subsequence that converges to a point in the set. If you are talking about subsets of Euclidean spaces, then, of course, a set is compact if and only if it is both closed and bounded.

I mentioned "complete set", it was definded as: a set where every Cauchy sequence is convergent, from Chapter 2 of W. Rudin's Principle of Mathematical Analysis .

And "compact set" is definded as: every open cover of the set has a finite subcover. Thus the compact set I was talking about is of general difinition. And thanks for reminding me of the Euclidean one.

## What is the difference between a perfect set and a complete set?

A perfect set is a set that contains all of its limit points, while a complete set is a set that contains all of its Cauchy sequences.

## How do you compare a perfect set and a complete set?

To compare a perfect set and a complete set, we can look at their definitions and determine if they satisfy the necessary conditions. A perfect set must contain all of its limit points, while a complete set must contain all of its Cauchy sequences. If a set satisfies both of these conditions, it can be considered both perfect and complete.

## What is the significance of verifying a compact set?

Verifying a compact set is important because it ensures that the set satisfies the necessary conditions for compactness. A compact set is a set that is both closed and bounded, and verifying this property can help us determine if a set is suitable for certain mathematical operations and proofs.

## Can a perfect set be compact but not complete?

Yes, a perfect set can be compact but not complete. An example of this is the set of rational numbers in the interval [0,1]. This set is perfect because it contains all of its limit points, but it is not complete because it does not contain all of its Cauchy sequences.

## What are the practical applications of comparing perfect sets and complete sets?

Comparing perfect sets and complete sets can be useful in various fields of mathematics, such as analysis and topology. It can also be applied in physics and engineering, where sets are used to model and describe real-world phenomena. Understanding the properties and differences between perfect sets and complete sets can help in solving problems and making accurate predictions in these areas.

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