What Does The Notation of Union Over K represent?
Authors often leave out the domain of the union if it is clear within the context. Here, $k$ is a natural number, i.e. $P = \bigcup\limits_{k \in \mathbb{N}} TIME(n^k) $. This is what it means for a complexity function to be bounded by a polynomial: there is a constant $k$ such that the function is in $\mathcal{O}(n^k)$.