embedding a T1 space into a product topology is the following statement true or false:
Let $X$ be a $T_1$ topological space, there exists a cofinite topological space $Y$, such that $X$ can be embedded into ${Y}^J$

Here's an easy proof that you can always do this with $|Y|=|X|$.  For any closed set $C\subset X$ and $x\in X\setminus C$, define a map $f:X\to Y$ which maps all of $C$ to a single point and is otherwise injective (we can do this since $|Y|\geq |X|$).  If $X$ is $T_1$ and $Y$ has the cofinite topology, this $f$ will be continuous, since the inverse image of each point is closed.  Moreover, $f(x)\not\in\overline{f(C)}$.  That is, continuous functions $X\to Y$ separate points from closed sets.  It follows that if you take $J$ to be the set of all continuous maps $X\to Y$  and let $F:X\to Y^J$ be the canonical map, then $F$ is an embedding.
