Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ times}}\;e_4\;\underbrace{00\dots00}_{4!\text{ times}}\;e_5\;\underbrace{00\dots00}_{5!\text{ times}}\dots\\ $$ where $e_i\in\{1,2\}$.
I found this set when I was trying to find a counter-example for the following statement
"Every uncountable set $A\subseteq\mathbb R$, has a rational limit point."
Obviously $K$ is uncountable. Now let $L$ denote the set of all Liouville numbers, then we can easily prove that $K\subset L$. But we know that the set $L$ is null and every element of $L$ is transcendental, hence $K$ is a null set and every element of $K$ is transcendental.
I proved that $K$ is a perfect set, i.e. $K'=K$, hence $K$ is a Baire space.
This set has many interesting properties but I just tried to write those with higher importance...for example every element of $K$ has unique decimal representation. You may find many interesting properties of this set.
I want to make this set interesting for others by showing that $K$ is a nice counter-example for many famous problems in mathematics. Thanks in advance...