Existence of well-filterifications of $T_0$ topological spaces

We prove that for every $T_0$ space $X$, there is a well-filtered space $W(X)$ and a continuous mapping $eta_X: Xlra W(X)$ such that for any well-filtered space $Y$ and any continuous mapping $f: Xlra Y$ there is a unique continuous mapping $hat{f}: W(X)lra Y$ such that $f=hat{f}circ eta_X$. Such a space $W(X)$ will be called the well-filterification of $X$. This result gives a positive answer to one of the major open problems on well-filtered spaces. Another result on well-filtered spaces we will prove is that the product of two well-filtered spaces is well-filtered.