I’ve added chapters “Cartesian closedness” and “Singularities” (from the site http://tiddlyspace.com which will be closed soon) to volume 2 draft.

Both chapters are very rough draft and present not rigorous proofs but rough ideas.

The journal European Journal of Pure and Applied Mathematics has accepted my
article after a peer review and asked me to send it in their LaTeX format.

I had a hyperref trouble with my LaTeX file. So I’ve said them that I
withdraw my article.

But later I realized that the best thing I can do is to remove hyperref
package.

After this I successfully converted my article to use their LaTeX package
and send it to European Journal of Pure and Applied Mathematics again. They
then said me that will publish the paper “this year”.

A few years passed.

It is yet unpublished and the editors of the journal ignore my emails, where
I remind them that they agreed to publish my article but don’t publish it.

What to do?

Note that my article is available online (the submitted version has a
different LaTeX template and some minor changes):

http://www.mathematics21.org/binaries/funcoids-reloids.pdf

I am reading the book “Convergence Foundations of Topology” by Szymon Dolecki and Frédéric Mynard. (Well, I am more skimming than reading, I may read it more carefully in the future.)

After reading about a half of the book, I tried to integrate my theory of funcoids with their theory of convergences.

And I noticed, that if I define convergences induced by funcoids following “Convergence of funcoids” chapter in my book, then convergences induced by (reflexive) funcoids are pretopologies (a narrow subclass of convergences). So convergences appear to be not a special case of funcoids. Certainly it seems that in the other direction funcoids are not a special case of convergences.

So we have a (probably difficult) problem: Find a common generalization of funcoids and convergences!

I (with some twist) described the set of $C^1$ integral curves for a given vector field in purely topological terms (well, I describe it not in terms of topological spaces, but in terms of funcoids, more abstract objects than topological spaces).

From this PDF file:

Theorem $f$ is a reparametrized integral curve for a direction field $d$ iff $f\in\mathrm{C}(\iota_D|\mathbb{R}|_{>};Q_+)\cap\mathrm{C}(\iota_D|\mathbb{R}|_{<};Q_-)$.

(Here $Q_+$ and $Q_-$ are certain funcoids determined by the vector field.)

You can understand this theorem after reading my research monograph.

I have re-defined filter rebase. Now it is defined for arbitrary filter $\mathcal{A}$ on some set $\mathrm{Base}(\mathcal{A})$ and arbitrary set $A$.

The new definition is: $\mathcal{A}\div A = \{ X\in\mathscr{P}A \mid \exists Y\in\mathcal{A}: Y\cap A\subseteq X \}$.

It is shown that for the special case of $\forall X\in\mathcal{A}:X\subseteq A$ the new definition is equal to the old definition that is $\mathcal{A}\div A = \{ X\in\mathscr{P}A \mid \exists Y\in\mathcal{A}: Y\subseteq X \}$.

See my book (updated), chapter “Orderings of filters in terms of reloids”, for details.

The new definition is useful for studying restrictions and embeddings of funcoids and reloids.

I have researched relations between directed topological spaces and pair of funcoids. Here the first funcoid represents topology and the second one represents direction.

Results are mainly negative:

Not every directed topological space can be represented as a pair of funcoids.

Different pairs of a topological space and its subfuncoid may generate the same directed topological space.

Conjecture Let $R$ be the complete funcoid corresponding to the usual topology on extended real line $[-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\}$. Let $\geq$ be the order on this set. Then $R\sqcap^{\mathsf{FCD}}\mathord{\geq}$ is a complete funcoid.