Closed and Open Electromagnetic Cores I think I may understand this concept, but I haven't been able to verify with anyone that I'm going about this the right way.
So when a wire is wrapped around a core that is a closed loop, like a large iron ring, and current passes through the wire, an electromagnetic field is created that goes through the iron core. However, according to my textbook, it says that when there is an opening in the core, the reluctance is greater because the air that the flux lines must go through is less permeable. It also says that the closer the north and south poles are to one another, the reluctance is reduced.
Now this tells me that (a) the magnetic flux does not interact with any external object if the core is a closed loop because the flux is isolated within the core, and (b) that an open core, like a horseshoe-shaped iron bar, is able to attract metal objects because it is an open loop, therefore the flux lines pour out of one end and go toward the other. Therefore, any object that approaches the poles will be attracted because they are much more permeable than air, hence will make a better path for the magnetic flux. Is this correct, or am I missing/misunderstanding one or a few things?
 A: This looks pretty much correct to me, except that it is not true that 

any object that approaches the poles will be attracted because they are much more permeable than air

Any object that is ferromagnetic will be very strongly attracted.
Paramagnetic materials may also be attracted, but less strongly.
diamagnetic matrials would be repelled.
An interesting example is that oxygen is paramagnetic - as proven by experiments such as the one shown here

where liquid oxygen preferentially fills the gap between N and S poles of a magnet. Nitrogen is diamagnetic and does not behave in the same way - it would simply pass the magnet.
To see the permeability of some common materials see here - the relative permeability of diamagnetic and paramagnetic materials is $\sim 1$, but for diamagnetic materials it is less than $1$ and for paramagetic materials it is more than $1$ - Ferromagnetic materials can have very large relative permeabilities of hundreds or thousands or more. Ferromagnetic materials also have the property that they can be magnetized and become permanent magnets.
