The discussion clarifies the distinctions between "perfect sets" and "complete sets" in the context of topology and mathematical analysis. A perfect set is defined as a non-empty, closed set with no isolated points. In contrast, the term "complete set" is not commonly recognized in topology but is defined in analysis as a set where every Cauchy sequence converges. Additionally, a compact set is characterized by the property that every open cover has a finite subcover, and in Euclidean spaces, it is both closed and bounded.
PREREQUISITESMathematicians, students of topology and analysis, and anyone interested in understanding the properties of sets in mathematical contexts.
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!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?
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.2. How can I verify whether a set is compact or not more evidently and effective?
Tks!