Does every uncountable Borel subset of $\mathbb R$ contains a perfect subset？

Yes - this is called the perfect set property. The analytic (= continuous image of Borel) sets also have this property; for more complicated sets (e.g. complements of analytic sets), the perfect set question is undecidable from the usual axioms of set theory.

While it's trivially true for open sets, already for closed sets it takes some work (the easiest approach, given a closed set $C$, is to consider the set of elements of $C$ around which $C$ is "locally uncountable" - that is, those $c\in C$ such that every open $U$ containing $c$ has uncountable intersection with $C$).

The full result follows from Borel determinacy, which is a very hard theorem; off the top of my head I don't know a proof that doesn't use this. Kechris' book Descriptive set theory is a very good source (as is Moschovakis' book).

Tags:

Set Theory

Borel Sets

Descriptive Set Theory

Related

About the construction of the restriction of a Vector Bundle in John M. Lee's book Show that $\int_{-\infty}^{\infty}f(x)\overline {g(x)}dx = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\mu)\overline{\hat{g}(\mu)}d\mu.$ Find Jordan Decomposition of $\left(\begin{smallmatrix} 4 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 3 \end{smallmatrix}\right)$ over $\mathbb{F}_5$ Closed-forms for the integral $\int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$? Quotient of two integrals $\frac{\int_0^\pi x^3\ln(\sin x)\,dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x))\,dx}$ How do the consequences of Russell's paradox extend beyond universal comprehension principle as far as the set of all sets problem? Is it possible to calculate consecutive digits of $\pi^{1/\pi}$ How prove this identity wih the sum equal to other sum What's the probability that the teacher teaches her class? If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0.$ Why is this integration method not valid? A nasty indefinite integral $ \int \frac{1}{x(x+1)(x+2)(x+3)(x+4) ... (x+m)} dx $

Recent Posts