I present my mathematical theory of singularities. It may probably have applications in general relativity and other physics.

The definitions are presented in this short draft article.

Before reading this article I recommend to skim through my research monograph (in the field of general topology), because the above mentioned article uses concepts defined in my book.

In short: I have defined “meta-singular numbers” which extend customary (real, complex, vector, etc.) numbers with values which functions take at singularities. If we allow meta-singular solutions of (partial) differential equations (such as general relativity) the equations remain the same, but the meaning of them changes. As such, we may probably get a modified version of general relativity and other theories.

Please collaborate with me to apply my theory to general relativity (I am no expert in relativity) and share half of Nobel Prize with me (if the results will be interesting, what I don’t 100% warrant now).

Anyway, we now have an interesting topic of research: What’s about solutions of differential equations in terms of meta-singular numbers?

I’ve added to my book two following theorems (formerly conjectures).

Theorem Let $\mu$ and $\nu$ are endoreloids. Let $f$ is a principal $\mathrm{C}' ( \mu; \nu)$ continuous, monovalued, surjective reloid. Then if $\mu$ is $\beta$-totally bounded then $\nu$ is also $\beta$-totally bounded.

Theorem Let $\mu$ and $\nu$ are endoreloids. Let $f$ is a principal $\mathrm{C}'' (\mu ; \nu)$ continuous, surjective reloid. Then if $\mu$ is $\alpha$-totally bounded then $\nu$ is also $\alpha$-totally bounded.

Reloid is a triple ${( A ; B ; F)}$ where ${A}$, ${B}$ are sets and ${F}$ is a filter on their cartesian product ${A \times B}$.

Endoreloid is reloid with the same ${A}$ and ${B}$.

Uniform space is essentially a special case of an endoreloid.

The reverse reloid is defined by the formula ${( A ; B ; F)^{- 1} = ( B ; A ; F^{- 1})}$.

See here about the definition of composition of reloids.

I define partial order on the set of reloids as ${( A ; B ; F) \sqsubseteq ( A ; B ; G) \Leftrightarrow F \supseteq G}$. The set of reloids with given source and destination is a complete lattice with join denoted ${\sqcup}$ and meet denotes ${\sqcap}$.

Uniform space is essentially the same as symmetric (${f = f^{- 1}}$), reflexive (every entourage contains the diagonal), and transitive ${( f \circ f \sqsubseteq f)}$ endo-reloid.

Traditionally the definition of quasi-uniform space is formed removing symmetry axiom. Other axioms including transitivity ${( f \circ f \sqsubseteq f)}$ remain unchanged.

I propose an alternate (non equivalent) definition of quasi-uniform spaces: Remove symmetry axiom and replace ${f \circ f \sqsubseteq f}$ with ${f \circ f^{- 1} \sqsubseteq f}$.

Why? Because ${f \circ f^{- 1} \sqsubseteq f}$ (not ${f \circ f = f}$) is a condition used in an important theorem (see this short article and also my book):

Theorem 1 Let ${f}$ is such a endoreloid that ${f \circ f^{- 1} \sqsubseteq f}$. Then ${f}$is ${\alpha}$-totally bounded iff it is ${\beta}$-totally bounded.

If we change the definition of quasi-unifrom spaces, this theorem applies to every quasi-uniform space. But if we follow the historical definition, it seems that this theorem does not work for every quasi-uniform space.

Now we should also change the definition of quasi-metric space to match the definition of quasi-uniform space, so that every quasi-metric induces a quasi-uniform space.

Metric spaces are defined by the axioms:

• ${d ( x ; y) \geqslant 0}$
• ${d ( x ; y) = 0 \Leftrightarrow x = y}$
• ${d ( x ; y) = d ( y ; x)}$ (symmetry)
• ${d ( x ; z) \leqslant d ( x ; y) + d ( y ; z)}$

The quasi-metric spaces are metric spaces without symmetry axiom.

I propose modified quasi-metric space with also

$\displaystyle d ( x ; z) \leqslant d ( x ; y) + d ( y ; z)$

replaced with

$\displaystyle d ( x ; z) \leqslant d ( y ; x) + d ( y ; z)$

(this is the same in presence of symmetry axiom).

Proposition 2 For this modified definition of quasi-metric spaces every quasi-metric induces my modified quasi-uniformity.

Proof: We need to prove that for every entourage ${U}$ there is an entourage ${V}$ such that ${V \circ V^{- 1} \subseteq U}$. Really, let ${U = \left\{ ( x ; y) \hspace{1em} | \hspace{1em} d ( x ; y) \leqslant \varepsilon \right\}}$ for some ${\varepsilon > 0}$. Then take ${V = \left\{ ( x ; y) \hspace{1em} | \hspace{1em} d ( x ; y) \leqslant \varepsilon / 2 \right\}}$ and thus we have

$\displaystyle V \circ V^{- 1} = \left\{ ( y ; z) \hspace{1em} | \hspace{1em} d ( y ; z) \leqslant \varepsilon / 2 \right\} \circ \left\{ ( x ; y) \hspace{1em} | \hspace{1em} d ( y ; x) \leqslant \varepsilon / 2 \right\} = \\ \left\{ ( x ; z) \hspace{1em} | \hspace{1em} \exists y : ( d ( y ; z) \leqslant \varepsilon / 2 \wedge d ( y ; x) \leqslant \varepsilon / 2) \right\} \subseteq \left\{ ( x ; z) \hspace{1em} | \hspace{1em} d ( x ; z) \leqslant \varepsilon \right\} = U$

because ${d ( y ; z) \leqslant \varepsilon / 2 \wedge d ( y ; x) \leqslant \varepsilon / 2 \Rightarrow d ( x ; z) \leqslant \varepsilon}$. $\Box$

The exact terms naming for new modified definitions of quasi-uniform an quasi-metric spaces is open for proposals by mathematical community.

Or maybe there are situations when the traditional definitions are better than mine? If so please comment on this blog post with your alternate opinion.

I’ve added new chapter 11 “Total boundness of reloids” to my book “Algebraic General Topology. Volume 1″.

It expresses several kinds of boundness of reloids, which are however the same total boundness in the special case of uniform spaces.

I realized that the terms “discrete funcoid” and “discrete reloid” conflict with conventional usage of “discrete topology” and “discrete uniformity”.

Thus I have renamed them into “principal funcoid” and “principal reloid”.

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).

Please post comments about properties of totally bounded spaces, in order to develop the theory further.

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 $\left\langle \left( \prod^{(S)}f \right)^{-1} \right\rangle y = \lambda i \in \mathrm{dom}\, f:\langle f_i^{-1} \rangle y_i$.