[I found that my computations below are erroneous, namely in general (the equality holds when is a set).]
During preparation my preprint of “Funcoids and Reloids” article I a little twisted on the problem of defining compact funcoids which was laying in my drafts.
I propose the following definition of compact funcoids: a funcoid is compact iff (in prose: the image of completion of the funcoid is the same as the image of the funcoid). I yet don’t know whether this is the best use of the word compact in the theory of funcoids. Maybe (for example) we are to call a funcoid compact if both and conform to this formula.
Now I will show how I obtained this formula:
My first attempt to define compactness of funcoids was the following formula which generalizes compactness defined by an addict of fido7.ru.math Victor Petrov (sadly I don’t remember the exact formula given by Victor Petrov and cannot find the original Usenet post):
Let’s equivalently transform this formula:
what is finally equivalent to the formula from the above: .
The most important thing we yet need to know about compact funcoids is the generalization of the theorem saying that to a compact topology corresponds unique uniformity. The generalization may sound like: to a compact funcoid corresponds a unique reloid. I have not yet found how to formulate exactly and prove this generalized theorem. Your comments are welcome.