Is there a set non-countable around zero but countable elsewhere?
Is there a set $A$ such that for every $a,r>0$ we have that $(0,r)\cap A$ is non-countable but $(a,a+r)\cap A$ is countable?
Solution 1:
No. Because $$(0,r) \cap A = \left( \bigcup_{n>0} \left(\frac{1}{n}, r\right) \right) \cap A = \bigcup_{n>0} \left(\left(\frac{1}{n}, r\right) \cap A \right)$$
But a countable union of countable set is countable, hence if every $(\frac{1}{n}, r) \cap A$ is countable, $(0,r)\cap A$ is countable