Counterexample for P being a closed set with isolated points in real analysis

In summary, for the given statement, "If P is the set of all isolated points of S, then P is a closed set," the concept of isolated points means that the set of isolated points is far apart from the rest of the points in the set. However, it is possible to find a counterexample, such as a sequence of numbers that converge to a limit, where every point is an isolated point. Therefore, the given statement is not always true.
  • #1
Valerie
1
0
For my homework, I have to find a counterexample for this: (with S being a subset of the reals.)
If P is the set of all isolated points of S, then P is a closed set.

I don't quite understand the concept of isolated points, which might be why I can't figure out a counterexample.
 
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  • #3
If it > 0 is any real number and x < y, show that there exists a rational number r such that
x < ru < y. (Hence the set (ru: r e Q) is dense in R.)
 
  • #4
Valerie said:
For my homework, I have to find a counterexample for this: (with S being a subset of the reals.)
If P is the set of all isolated points of S, then P is a closed set.

I don't quite understand the concept of isolated points, which might be why I can't figure out a counterexample.

An "isolated" is just a point that is "away" from other points in the set. More technically, there is some neighborhood (open set) containing the point that does not contain any other point of the set. For example, [itex][0, 1]\cup {2}[/itex], the set of all numbers between 0 and 1 (inclusive) and the number 2, has "2" as an isolated point. If a set is a sequence, say, {1, 2, 3, 4, 5}, then every point is an isolated point.

Think about a sequence of numbers that converge to some limit. What are its isolated points?
 

What is Real Analysis?

Real Analysis is a branch of mathematics that deals with the properties and operations of real numbers. It involves the study of functions, sequences, limits, continuity, differentiation, and integration.

What are the applications of Real Analysis?

Real Analysis has numerous applications in various fields such as physics, engineering, economics, and computer science. It is used to model and analyze real-world problems and systems, and to make predictions and decisions based on mathematical principles.

What are the key concepts in Real Analysis?

Some of the key concepts in Real Analysis include sets, numbers, functions, limits, continuity, differentiation, integration, and series. These concepts are used to understand and analyze the behavior of real-valued functions and their properties.

What are some common techniques used in solving Real Analysis problems?

Some common techniques used in solving Real Analysis problems include the use of theorems and definitions, proofs, theorems and their converses, counterexamples, and the application of various mathematical tools and methods such as calculus, algebra, and topology.

What are some tips for approaching Real Analysis problems?

Some helpful tips for approaching Real Analysis problems include understanding the definitions and theorems, practicing with examples, breaking down complex problems into smaller parts, and using various techniques and approaches to solve them. It is also important to stay organized, logical, and patient while solving Real Analysis problems.

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