Comparing PERFECT SET vs COMPLETE SET | Verifying Compact Sets

[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!

 

[HallsofIvy]

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.

 

[Ka Yan]

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.