Let $A$ be an uncountable set of real numbers. Then by using bolzano-weierstrass theorem, it can be easily shown that $A$ has at least one limit point. But my question is how many limit point? Can an uncountable set has countable number of limit points? Please suggest me. Thanks in advance.
[Math] Limit points of an Uncountable set of real numbers.
real-analysis
Best Answer
Without loss of generality, we can assume $A$ is closed, since the limit points of $\overline{A}$ are exactly the limit points of $A$. The Cantor-Bendixson Theorem says that every uncountable, closed subset of the reals contains a nonempty perfect subset. (A perfect set is a set with no isolated points, i.e., a set where every point is a limit point.) It is a fact that every (nonempty) perfect subset of $\mathbb{R}$ has cardinality $2^{\aleph_0}$ (proof idea: build a tree of closed sets, and use Bolzanno-Weierstrass to show that every path along the tree contains a distinct point), $A$ has at least $2^{\aleph_0}$ limit points. Since there are only $2^{\aleph_0}$ real numbers to begin with, this answer is exact.