I proved:

Theorem $T$ is a left adjoint of both $F_{\star}$ and $F^{\star}$, with bijection which preserves the “function” part of the morphism.

The details and the proof is available in the draft of second volume of my online book.

The proof is not yet enough checked for errors.

After proposing this conjecture I quickly found a counterexample:

$S = \left\{ (- a ; a) \mid a \in \mathbb{R}, 0 < a < 1 \right\}$, $f$ is the usual Kuratowski closure for $\mathbb{R}$.

Conjecture $\langle f \rangle \bigsqcup S = \bigsqcup_{\mathcal{X} \in S} \langle f \rangle \mathcal{X}$ if $S$ is a totally ordered (generalize for a filter base) set of filters (or at least set of sets).

I started research of mappings between endofuncoids and topological spaces.

Currently the draft is located in volume 2 draft of my online book.

I define mappings back and forth between endofuncoids and topologies.

The main result is a representation of an endofuncoid induced by a topological space.

The formula is $f\mapsto 1\sqcup\mathrm{Compl}\, f\sqcup(\mathrm{Compl}\, f)^2\sqcup \dots$.

However I proved this theorem only for the special case if every singleton is a closed set. Also the proof is not yet checked for errors.

I have added to my book section “Expressing limits as implications”.

The main (easy to prove) theorem basically states that $\lim_{x\to\alpha} f(x) = \beta$ when $x\to\alpha$ implies $f(x)\to\beta$. Here $x$ can be taken an arbitrary filter or just arbitrary ultrafilter.

The section also contains another, a little less obvious theorem. There is also a (seemingly easy) open problem there.

This is partly an offtopic post in my math blog.

It seems likely that I discovered a category in which such objects as the Father and the Son from the Gospel appear. I am not sure I really discovered God, but this seems likely.

Consider a category (there seems to be multiple ways to add morphisms, so I will speak only about objects, not morphisms) whose objects are $(X,Y,I,Z)$ where $X\supseteq Y$$I$ are sets and $Z:X\times I\to Y$ is a function of two arguments. $X$ is “the Father” (argument of the function $Z$) and $Y$ is “the Son” (the result of the function $Z$). That is “Father” means “argument” and “Son” means “result”. $Z$ is a function which “proceeds” (see John 15:26) from the Father and so likely is Holy Spirit.

This category is somehow powerful. For example one of its objects is the system of all propositional (or if you like, predicate) formulas. So it is probably “big enough” to entail everything about mathematics and everything we can know about God.

Read my preliminary drafts of this theory at this page where properties of such “systems of formulas” are studied in more details.

In 2005 year I put online some math articles related with formulas and math logic (despite I am not a professional logician).

In 2005 I like a crackpot thought that I discovered a completely new math method replacing axiomatic method. This was a huge error (my skipped proof was just wrong).

After that the files were forgotten.

Nevertheless I remembered about my old articles and hope they may be useful after clearing from the hype. (I have removed fragments about being a huge discovery.)

The articles are now available here:

I my old files there was also written that X (“argument”) is Father and Y (“result”) in my writings is Son (as of the Gospel). “Index” probably is Holy Spirit. That is I probably discovered the topic of which Gospels talks.

However I do not remember details of this my old theory and need to re-read it myself to understand again.