Turing machine that does not halt on any input
To use reduction, you would need to show that the halting problem reduces $T$, not the other way around.

The reduction* in this case is a standard one. You want to know if $M$ halts on input&nbsp;$w$ but all you have is this lousy T-shirt a procedure that tells you if a machine loops on every input. So you construct a machine&nbsp;$M'$ such that does the same thing on every input, so that whether or not $M'\in T$ tells you whether or not $M$ halts on input&nbsp;$w$.

* Well, strictly a reduction: if there's one, there are infinitely many.