Unimpeachable fact: one can construct both inductors & transformers without any permeable core material whatsoever!

Yes that are facts well known for any student fail making at exams, if due, ask student where resides the energy at capacitor, and where at coil and highly probable he will fail. (Its from my tutoring practice how to revenge with a glance). BUT, the sorrow practice is bound with the question - if the core-less transformer may economy a multi-tonn weighting core, then whay all the transformers are not gonna made in that way? And answer may make You sorrow - the steel suppresss the turns cound some 4400 fold. And if insteas of say 10 000 turns one must wound a 4 4000 000 turns, the bobbins will be so giant that price (yes price) of copper will become unbearable.
Thus, is no matter where the field resides, the money is all the worth, and money minimization demands the use of core, except the really radiofrequency transformers, where number of counts become more or less normal (and even then not always - antenna baluns rarely are ferrite-less).

 

Ferrite is not as good of a low frequency transformer material as the steel cores are. >>

In the pure theory, low freq is nothing inherently wrong for ferrites. Just it have some 4 to 6 fold lesser saturation induction (Tesla count) therefore demand the 4 to 6 fold larger core than steel. Yet the ferrite weight is sure lesser than steel, but most often the size is limiting any appliance not the wight per se. That was reason one. Reason two is that identical size of core cost many much more at case of ferrites into comparizon with steel.

By the way, waving to Indians, one certain India company with US governing body is producing in India ferrite I cores in size of 1x1x4 inch. It have enviable electric parameters of 0.36 Tesla saturation and 3000 of mju, them are easily gluable together forming the ANY shape and size of ferrite megatransformers (E-type, O-type etc) and only the fault is rather fragile brittleness, just must move it by soft hands. The last but no least, the price for one I element is something between 1 and 2 Euro depending on stock size. We bought it from Checzk mediator, and are satisfied like a rabbit in between dandelion.
In the picture - my 120 kW, 30 kV, 5 A, 50 kHz ferrite transformer for ZVS SMPS I made some years ago. Core size 32 cm x 18 cm x 5 cm. Specific layer style - each layer have both ends out, and connected in the manner let the all layers have only one layer voltage between any neighbor layer point. Normal winding have some points about zero, and some points double one layer voltage. I was looking to minimize the beat-through effect at 5 kV per layer.