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March 19, 2010 / porton

On the definition of compact funcoids

[I found that my computations below are erroneous, namely \mathrm{Cor} \langle f^{-1}\rangle \mathcal{F} \neq \langle \mathrm{CoCompl} f^{-1}\rangle  \mathcal{F} in general (the equality holds when \mathcal{F} 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 f is compact iff \mathrm{im} f = \mathrm{im} \mathrm{Compl} f (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 f compact if both f and f^{-1} 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):

\forall \mathcal{F} \in \mathfrak{F} : ( \langle f^{-1}\rangle \mathcal{F} \neq \emptyset \Rightarrow \exists \alpha : \{\alpha\} \subseteq \langle f^{-1}\rangle  \mathcal{F}).

Let’s equivalently transform this formula:

\forall \mathcal{F} \in \mathfrak{F} : ( \langle f^{-1}\rangle \mathcal{F} \neq \emptyset \Rightarrow \mathrm{Cor} \langle f^{-1}\rangle  \mathcal{F} \neq \emptyset);

\forall \mathcal{F} \in \mathfrak{F} : ( \langle f^{-1}\rangle \mathcal{F} \neq \emptyset \Rightarrow \langle \mathrm{CoCompl} f^{-1}\rangle  \mathcal{F} \neq \emptyset);

\forall \mathcal{F} \in \mathfrak{F} : ( \mathcal{F} \cap^{\mathfrak{F}} \mathrm{dom} f^{-1}\neq \emptyset \Rightarrow \mathcal{F} \cap^{\mathfrak{F}} \mathrm{dom} \mathrm{CoCompl} f^{-1}\neq \emptyset);

\mathrm{dom} f^{-1} = \mathrm{dom} \mathrm{CoCompl} f^{-1} what is finally equivalent to the formula from the above: \mathrm{im} f = \mathrm{im} \mathrm{Compl} f.

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.

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