Is the unweighted vertex cover problem equivalent to its weighted version?
If you're talking about determining whether there exists a cover of weight $\le T$, then the problem is obviously in NP and thus reduces to unweighted vc (since $VC$ is NP complete). 

If however were talking about whether the minimal cover is of weight $\le T$, then the problem is $\Sigma_2$ complete, and is therefore not reducible to vc unless the hierarchy collapses to the first level.