Converting between $T_1$ and $T_2$ Given that every $T_2$ space is a $T_1$ space, is it possible to start with a $T_1$ space and to specify in terms of its sets a family of additional sets sufficient to make that $T_1$ space into a $T_2$ space? If so, can this be done for any $T_1$ space or only particular ones?

I suspect that every way of doing this is somehow either arbitrary or trivial.
Suppose we have an operation $\mathcal{H}$, such that if $(X, T)$ is a T1 space
then $T \subset \mathcal{H}(T)$ and $(X, \mathcal{H}(T))$ is a T2 space.
Suppose furthermore that this operation preserves symmetry, in the sense that if
$f: X \to X$ is a homeomorphism of $(X, T)$ to itself, then it is also a homeomorphism of
$(X, \mathcal{H}(T))$ to itself.
Finally, suppose that the operation is monotone, in the sense that if $T \subset T'$, then
$\mathcal{H}(T) \subset \mathcal{H}(T')$.
With these, apparently reasonable, assumptions it is already unavoidable that $(X, \mathcal{H}(T))$
is discrete, whatever $T$ is. To see this, consider the cofinite topology. It is T1
and every bijection is a homeomorphism. The only T2 topology with the property
that every bijection is a homeomorphism is the discrete one.  Since the cofinite topology
is the coarsest T1 topology and the discrete topology is the finest of all,
monotonicity then demands that $\mathcal{H}$ makes every T1 topology discrete.
