Here I present my further research plans in Algebraic General Topology which is a new field of math developed by me. This research plan is not formal and may contain vague statements.

## Open problems

First, there are some open problems (conjectures) in AGT [HTML]

(PDF). These should be solved. Most of these problems are also available in Open Problems Garden for discussion.

Probably Stone duality can be used to transfer some of these open problems into problems about topological spaces, what may help to solve them.

## Category theory

Misc properties of continuous functions between endofuncoids and endoreloids.

(A;B)->FCD(A;B) for posets A, B is a tensor product (in category theory sense). It is specifically likely that it is a tensor product in the category of join-semilattices or a similar category. See here for a proof.

## Compact funcoids

Generalize the theorem that compact topology corresponds to only one uniformity.

For compact funcoids the Cantor’s theorem that a function continuous on a compact is uniformly continuous.

Every closed subset of a compact space is compact. A compact subset of a Hausdorff space is closed. 17.5 theorem in Willard.

17.6 theorem in Willard.

17.7 theorem in Willard: The continuous image of a compact space is compact.

17.10 Theorem in Willard: A compact Hausdorff space X is a T_4-space. Also 17.11 Corollary, 17.13, 17.14 theorem.

“Locally compact” for funcoids. See also 18 “Locallyγ compact spaces: in Willard.

Compactification

## Pointless topology

Algebraic General Topology which is currently a field of point-set topology should be generalized to pointless topology:

After this we can define reloid on the lattice as a filter on the set of relations between lattice elements.

Or maybe it is equivalent to defining reloids as filters on the set of funcoids – this conjecture needs to be further researched (first formalized then solved).

Consider pointfree reloids defined as RLD(A;B) = RLD(atoms^{A}; atoms^{B}) for posets A, B. What are their relationships with pointfree funcoids?

Pointfree reloids are finally defined in this blog post.

## Multidimensional funcoids

Research in more details.

## Misc

Show that cross-composition product is a special case of infimum product.

Analog of order topology for funcoids/reloids.

If a morphism converges to a value on two sets, it converges to the value also on their union.

I expect that the following will be a common generalization of reloids and funcoids: a complete lattice filtrator whose base is the set of binary relations (on some set). However now is not enough known about the filtrator of funcoids, e.g. whether it is finitely meet-closed.

A set is connected if every function from it to a discrete space is constant. Can this be generalized for generalized connectedness and generalized continuity? I have no idea how to relate these two concepts in general.

Develop theory of *funcoidal groups* by analogy with topological groups. Attempt to use this theory to solve this open problem. Is it useful as topological group determines not only a topology but even a uniformity?

A space μ is T_2- iff the diagonal Δ is closed in μ×μ.

The β-th projection map is not only continuous but also open (Willard, theorem 8.6).

T_x-separation axioms for products of spaces.

Willard 13.13 and its important corollary 13.14.

Willard 15.10.

About real-valued functions on endofuncoids: Urysohn’s Lemma (and consequences: Tietze’s extension theorem) for funcoids.

See also this blog post.

Generalized Fréchet filter on a poset 𝔄 is a filter Ω such that ∂Ω={x∈𝔄 | atoms x is infinite}. Research their properties (first, whether they exists for every poset).

Manifolds.

“Micronization” of partial orders: this blog post.

this article (free download, also Google for “pre-adjunction”, also “semi” instead of “pre”) Are (FCD) and (RLD)_{in} adjunct?