The definition of the initial topology on Wikipedia (and in several books) is as follows:
Explicitly, the initial topology is the collection of open sets generated by all sets of the form $\displaystyle f_{i}^{-1}(U)$, where $U$ is an open set in $\displaystyle Y_{i} $ for some $ i ∈ I$, under finite intersections and arbitrary unions.
Now we have the product topology, which is defined by the initial topology with the canonical injection from the product space to its components.
By our definition above, the preimage of $U = (0,1) \subset \mathbb{R}$ should also contain $(0,1) \times [0,1]$, if we look at the Space $\mathbb{R^{2}}$. Since $f_{1}((0,1) \times [0,1]) = (0,1)$. But $(0,1) \times [0,1]$ is obviously not open.