This is a straightforward generalization of the customary definition of totally bounded sets on uniform spaces:

Definition Reloid $f$ is totally bounded iff for every $E \in \mathrm{GR}\, f$ there exists a finite cover $S$ of $\mathrm{Ob}\, f$ such that $\forall A \in S : A \times A \subseteq E$.

See here for definitions and notation.

I don’t know which interesting properties totally bounded spaces have (except of their connection to compact spaces, but at the time of writing this compactness of funcoids is not yet properly defined).

Today I’ve discovered a new kind of product of funcoids which I call “simple product”.

It is defined by the formulas
$\left\langle \prod^{(S)}f \right\rangle x = \lambda i \in \mathrm{dom}\, f: \langle f_i \rangle x_i$ and $\left\langle \left( \prod^{(S)}f \right)^{-1} \right\rangle y = \lambda i \in \mathrm{dom}\, f:\langle f_i^{-1} \rangle y_i$.

I’ve put a partial partial proof of “Every filter on a set can be strongly partitioned into ultrafilters” conjecture at PlanetMath. Please collaborate in solving this conjecture.

I’ve created a site where anyone may list his projects and anyone may mark which projects he is going to participate. Projects are organized into a tree. The site supports LaTeX and has “Mathematics” section: http://theses.portonvictor.org/node/2

I have posted several pages on my math research project: http://theses.portonvictor.org/node/4 – feel free to add yours also.

The site is ideal for posting possible theses topics for students as well as cutting edge research for math persons. For a topic a user may mark himself as an adviser, which could help students.

Among mathematics this site is also going to list open source software proposals and whatever general utility projects.

Well, this project needs its own domain name. Please somebody donate a domain and help with hosting.

I have proved that for every funcoid $f:\prod A\rightarrow\prod B$ (where $A$ and $B$ are indexed families of sets) there exists a funcoid $\mathrm{Pr}^{(A)}_k f$ (subatomic projection) defined by the formula:

$\mathcal{X} \mathrel{\left[ \Pr^{\left( A \right)}_k f \right]} \mathcal{Y} \Leftrightarrow \\ \prod^{\mathsf{RLD}}_{i \in \mathrm{dom}\, A} \left( \left\{ \begin{array}{ll} 1^{\mathfrak{F} \left( A_i \right)} & \mathrm{if}\, i \neq k ;\\ \mathcal{X} & \mathrm{if}\, i = k \end{array} \right. \right) \mathrel{\left[ f \right]} \prod^{\mathsf{RLD}}_{i \in \mathrm{dom}\, B} \left( \left\{ \begin{array}{ll} 1^{\mathfrak{F} \left( B_i \right)} & \mathrm{if}\, i \neq k ;\\ \mathcal{Y} & \mathrm{if}\, i = k \end{array} \right. \right) .$

My draft book is modified to include this new theorem.

I had a very great idea in the field of general topology, what I now call funcoids.

Years of my research of funcoids culminated me writing a research monograph about funcoids and related stuff.

But after I’ve finished this monograph and submitted it to a publisher, I realize that I get stuck. I have no idea how to prove any of conjectures I formulated, neither see any definitions of new concepts which could advance my research further.

I even think to switch to research in some other, unrelated field of mathematics.

This my book is probably already worth Abel Prize or Fields Medal, but I have no idea how to advance it further even for a little, I got stuck.

I added the definition and properties of “second reloidal product” (the definition was inspired by Tychonoff product of topological spaces) to my research monograph “Algebraic General Topology. Volume 1″.

See the subsection “Second reloidal product” in the section “Multireloids”.