micromass said:The construction of such a set is quite tricky. You will have to modify the construction of the Cantor set in such a way that it misses the rationals. See http://en.wikipedia.org/wiki/Smith–Volterra–Cantor_set for general information about such a set.
yes, you are right...sorry, I made a mistake in my proof... but now I can understand it...thank U very much...micromass said:Throw out a rational number at each step of the construction.
We should choose the end points of each segments carefully to make sure that the end points are all irrational numbers and the set is still closed!micromass said:Throw out a rational number at each step of the construction.
A perfect set is a set of real numbers that has the same cardinality as the set of all real numbers, and is also closed and uncountable.
The problem about perfect set is important because it helps us understand the properties and behavior of infinite sets, which have many applications in mathematics and other fields of science.
The perfect set problem is significant in mathematics because it is closely related to other important concepts such as topology, measure theory, and set theory. It also has connections to other famous mathematical problems, such as the Continuum Hypothesis.
No, the perfect set problem has not been solved. It remains an open problem in mathematics, and many mathematicians continue to work on finding a solution.
Some proposed solutions or approaches to the perfect set problem include the use of descriptive set theory, forcing, and large cardinals. However, none of these approaches have yet yielded a complete solution to the problem.