## Some new results on finite algebras whose only known proof uses large cardinals (updated)

The large cardinals above hugeness are a very powerful tool for proving results about finite self-distributive algebras, but mathematicians so far have neglected to exploit the tremendous power of these very large cardinals to prove results about finite objects. Initially around the late 80’s and early 90’s, Laver and other mathematicians have established some very good classical results about the Laver tables. On the other hand, from the later 1990’s to the 2010’s, no mathematician has published any original research that directly relates to what I call Laver-like LD-systems. Furthermore, before my work on the algebras of elementary embeddings, nobody has investigated what will happen in the algebras of elementary embeddings generated by more than one element.
There are two results about the classical Laver tables which presumably require large cardinals to establish. One of these results states that the inverse limit of the classical Laver tables contains free left-distributive algebras on one generator while the other result states that in the classical Laver table $A_{n}$, we have $2*_{n}x=2^{n}\Rightarrow 1*_{n}x=2^{n}$. All of the other results about the classical Laver tables are known to hold in ZFC.

In this post, we shall use large cardinals and forcing to prove the existence of certain classes of finite self-distributive algebras with a compatible linear ordering. The results contained in this note shall be included in my (hopefully soon to be on Arxiv) 100+ page paper Generalizations of Laver tables. In this post, I have made no attempt to optimize the large cardinal hypotheses.

For background information, see this post or see Chapter 11 in the Handbook of Set Theory.

We shall let $\mathcal{E}_{\alpha}$ denote the set of all elementary embeddings $j:V_{\alpha}\rightarrow V_{\alpha}.$
By this answer, I have outlined a proof that the algebra $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is locally finite. We therefore have established a deep connection between the top of the large cardinal hierarchy and finite algebras.

In this note, we shall use two important ideas to construct finite self-distributive algebras. The main idea is to generalize the square root lemma for elementary embeddings so that one obtains elementary embeddings with the desired properties.

$\textbf{Theorem: (Square Root Lemma)}$ Let $j\in\mathcal{E}_{\lambda+1}$. Then there is some $k\in\mathcal{E}_{\lambda}$ where $k*k=j|_{V_{\lambda}}$.

$\mathbf{Proof}:$ By elementarity
$$V_{\lambda+1}\models\exists k\in\mathcal{E}_{\lambda}:k*k=j|_{V_{\lambda}}$$
if and only if
$$V_{\lambda+1}\models\exists k\in\mathcal{E}_{\lambda}:k*k=j(j|_{V_{\lambda}})$$
which is true. Therefore, there is some $k\in\mathcal{E}_{\lambda}$ with $k*k=j|_{V_{\lambda}}$. $\mathbf{QED}$

The other idea is to work in a model such that there is a cardinal $\lambda$ where there are plenty of rank-into-rank embeddings from $V_{\lambda}$ to $V_{\lambda}$ but where $V_{\lambda}\models\text{V=HOD}$. If $V_{\lambda}\models\text{V=HOD}$, then $V_{\lambda}$ has a definable linear ordering which induces a desirable linear ordering on rank-into-rank embeddings and hence linear orderings on finite algebras. The following result can be found in this paper.

$\mathbf{Theorem}$ Suppose that there exists a non-trivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$. Then in some forcing extension $V[G]$ there is some elementary embedding $k:V[G]_{\lambda+1}\rightarrow V[G]_{\lambda+1}$ where
$V[G]_{\lambda}\models\text{V=HOD}$.

Therefore it is consistent relative to large cardinals that there exists a non-trivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ such that $V_{\lambda}\models\text{V=HOD}$.

Now suppose that $V_{\lambda}\models\text{V=HOD}$. Then there exists a linear ordering $\ll$ of $V_{\lambda}$ which is definable in $V_{\lambda}$. If $j\in\mathcal{E}_{\lambda}$ and $\gamma$ is a limit ordinal with $\gamma<\lambda$, then define $j\upharpoonright_{\gamma}:V_{\gamma}\rightarrow V_{\gamma+1}$ by $j\upharpoonright_{\gamma}(x)=x\cap V_{\gamma}$ for each $x\in V_{\gamma}.$ Take note that $j\upharpoonright_{\gamma}=k\upharpoonright_{\gamma}$ if and only if $j\equiv^{\gamma}k$. Define a linear ordering $\trianglelefteq$ on $\mathcal{E}_{\lambda}$ where $j\trianglelefteq k$ if and only if $j=k$ or there is a limit ordinal $\alpha$ where $j\upharpoonright_{\alpha}\ll k\upharpoonright_{\alpha}$ but where $j\upharpoonright_{\beta}=k\upharpoonright_{\beta}$ whenever $\beta<\alpha$. Define a linear ordering $\trianglelefteq$ on $\{j\upharpoonright_{\gamma}\mid j\in\mathcal{E}_{\lambda}\}$ by letting $j\upharpoonright_{\gamma}\triangleleft k\upharpoonright_{\gamma}$ if and only if there is some limit ordinal $\beta\leq\gamma$ where $j\upharpoonright_{\beta}\ll k\upharpoonright_{\beta}$ but where $j\upharpoonright_{\alpha}=k\upharpoonright_{\alpha}$ whenever $\alpha$ is a limit ordinal with $\alpha<\beta$. By elementarity, the linear ordering $\trianglelefteq$ satisfies the following compatibility property: if $k\upharpoonright_{\gamma}\trianglelefteq l\upharpoonright_{\gamma}$, then $(j*k)\upharpoonright_{\gamma}\trianglelefteq(j*l)\upharpoonright_{\gamma}$. We say that a linear ordering $\leq$ on a Laver-like LD-system $(X,*)$ is a compatible linear ordering if $y\leq z\Rightarrow x*y\leq x*z$. If $V_{\lambda}\models\text{V=HOD}$, then $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ has a compatible linear ordering defined by $[j]_{\gamma}\leq[k]_{\gamma}$ if and only if $j\upharpoonright_{V_{\gamma}}\trianglelefteq k\upharpoonright_{V_{\gamma}}$.

Using generalized Laver tables, we know that the set $\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$ has order-type $\omega$. Let $\text{crit}_{r}(j_{1},…,j_{n})$ be the $r$-th element of the set $$\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$$ ($\text{crit}_{0}(j_{1},…,j_{n})$ is the least element of $\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$). Let $T:\bigcup_{n\in\omega}\mathcal{E}_{\lambda}^{n}\rightarrow V_{\omega\cdot 2}$ be a mapping definable in $(V_{\lambda+1},\in)$ where $T(j_{1},…,j_{m})=T(k_{1},…,k_{n})$ if and only if $m=n$ and if $\gamma=\text{crit}_{r} (j_{1},…,j_{m})$ and $\delta=\text{crit}_{r}(k_{1},…,k_{n})$, then there is some isomorphism $\phi:\langle j_{1},…,j_{m}\rangle/\equiv^{\gamma}\rightarrow\langle k_{1},…,k_{n}\rangle/\equiv^{\delta}$ where $\phi([j_{i}]_{\gamma})=[k_{i}]_{\delta}$. We remark that if $T(j_{1},…,j_{m})=T(k_{1},…,k_{n})$, then the subspaces $\overline{\langle j_{1},…,j_{m}\rangle}$ and $\overline{\langle k_{1},…,k_{n}\rangle}$ of $\mathcal{E}_{\lambda}$ are homeomorphic by an isomorphism of algebras preserving $*,\circ$ ($\mathcal{E}_{\lambda}$ can be given a complete metric that induces a canonical uniformity on $\mathcal{E}_{\lambda}$).

The following technical result is a generalization of the Square-Root Lemma, and a simplified special case of the following results can be found in this answer that I gave.

$\mathbf{Theorem:}$ Suppose the following:

1. $\ell_{1},…,\ell_{p},j_{1},…,j_{m}\in\mathcal{E}_{\lambda}$ and $(k_{r,s})_{1\leq r\leq n,1\leq s\leq p}\in(\mathcal{E}_{\lambda})^{n\cdot p}$.
2. $\ell_{1},…,\ell_{p}$ are I1 embeddings.
3. $T(\ell_{i}*j_{1},…,\ell_{i}*j_{m},k_{1,i},…,k_{n,i})=x_{i}$ whenever $1\leq i\leq p$.
4. $v$ is a natural number.
5. there is some $\mu<\lambda$ where $\mu=\text{crit}_{v}(k_{1,1},...,k_{n,1})=\ldots=\text{crit}_{v}(k_{1,p},...,k_{n,p})$.
6. $k_{r,1}\equiv^{\mu}…\equiv^{\mu}k_{r,p}$ for $1\leq r\leq n$.
7. $\ell_{1}\equiv^{\mu+\omega}…\equiv^{\mu+\omega}\ell_{p}$.

Then there are $(w_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ in $\mathcal{E}_{\lambda}$ where

1. $T(j_{1},…,j_{m},w_{1,i},…,w_{n,i})=x_{i}$ for $1\leq i\leq p$,
2. there is some $\alpha<\lambda$ where $\text{crit}_{v}(w_{1,i},...,w_{n,i})=\alpha$ for $1\leq i\leq p$, and
3. $w_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}w_{r,p}$ for $1\leq r\leq n$.

$\mathbf{Proof:}$ For $1\leq i\leq p$, let $A_{i}$
$$=\{(w_{1}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})},…,w_{n}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})}): T(j_{1},…,j_{m},w_{1},…,w_{n})=x_{i}\}.$$
Then $\ell_{i}(A_{i})$
$$=\{(w_{1}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})},…,w_{n}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})}): T(\ell_{i}*j_{1},…,\ell_{i}*j_{m},w_{1},…,w_{n})=x_{i}\}.$$
Therefore,
$$(k_{1,i}\upharpoonright_{\mu},…,k_{n,i}\upharpoonright_{\mu})\in\ell_{i}(A_{i})$$ for $1\leq i\leq p$. Since
$k_{r,1}\upharpoonright_{\mu}=…=k_{r,p}\upharpoonright_{\mu}$, we have
$$(k_{1,1}\upharpoonright_{\mu},…,k_{n,1}\upharpoonright_{\mu})=…=(k_{1,p}\upharpoonright_{\mu},…,k_{n,p}\upharpoonright_{\mu}).$$
Therefore, let $$(\mathfrak{k}_{1},…,\mathfrak{k}_{n})=(k_{1,1}\upharpoonright_{\mu},…,k_{n,1}\upharpoonright_{\mu}).$$
Then
$$(\mathfrak{k}_{1},…,\mathfrak{k}_{n})\in\ell_{1}(A_{1})\cap…\ell_{p}(A_{p})\cap V_{\mu+\omega}$$
$$=\ell_{1}(A_{1})\cap…\cap\ell_{1}(A_{p})\cap V_{\mu+\omega}$$
$$\subseteq\ell_{1}(A_{1}\cap…\cap A_{p}).$$
Therefore, $A_{1}\cap…\cap A_{p}\neq\emptyset.$

Let $(\mathfrak{w}_{1},…,\mathfrak{w}_{n})\in A_{1}\cap…\cap A_{p}$. Then there are $(w_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ in $\mathcal{E}_{\lambda}$ where
$$(\mathfrak{w}_{1},…,\mathfrak{w}_{n})=(w_{1,i}\upharpoonright_{\text{crit}_{v}(w_{1,i},…,w_{n,i})},…,w_{n,i}\upharpoonright_{\text{crit}_{v}(w_{1,i},…,w_{n,i})})$$
and
$$T(j_{1},…,j_{m},w_{1,i},…,w_{n,i})=x_{i}$$
for $1\leq i\leq p.$ Therefore, there is some $\alpha<\lambda$ with $\text{crit}_{v}(w_{1,s},...,w_{n,s})=\alpha$ for $1\leq s\leq p$ and where $w_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}w_{r,p}$ for $1\leq r\leq n$. $\mathbf{QED}$

$\mathbf{Remark:}$ The above theorem can be generalized further by considering the classes of rank-into-rank embeddings
described in this paper.

$\mathbf{Theorem:}$ Suppose that there exists an I1 cardinal. Suppose furthermore that

1. $U_{1},…,U_{p},V_{1},…,V_{m}$ and $(W_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ are unary terms in the language with function symbols $*,\circ$,
2. $L$ is an $np+1$-ary term in the language with function symbols $*,\circ$,
3. $v$ is a natural number,
4. There is some classical Laver table $A_{N}$ where in $A_{N}$, we have
$\mu=\text{crit}_{v}(W_{1,1}(1),…,W_{n,1}(1))=…=\text{crit}_{v}(W_{1,p}(1),…,W_{n,p}(1))<\text{crit}(2^{n}).$
5. For all $N$, we have $W_{r,1}\equiv^{\mu}\ldots\equiv^{\mu}W_{r,p}(1)$ for $1\leq r\leq n$,
6. $U_{1}(1)\equiv^{\mu^{+}}\ldots\equiv^{\mu^{+}}U_{p}(1)$.

If $Y$ is a finite reduced Laver-like LD-system, then let $\approx$ be the relation on $Y^{<\omega}$ where $(x_{1},...,x_{m})\approx(y_{1},...,y_{n})$ if and only if $m=n$ and whenever $\langle x_{1},...,x_{m}\rangle$ and $\langle y_{1},...,y_{n}\rangle$ both have more than $v+1$ critical points, then there is an isomorphism $\iota:\langle x_{1},...,x_{m}\rangle/\equiv^{\text{crit}_{v}(x_{1},...,x_{m})}\rightarrow \langle y_{1},...,y_{n}\rangle/\equiv^{\text{crit}_{v}(y_{1},...,y_{n})}$ where $\iota([x_{i}])=[y_{i}]$ for $1\leq i\leq n$.
Then there is some finite reduced Laver-like LD-system $X$ along with
$x,(y_{r,s})_{1\leq r\leq n,1\leq s\leq p}\in X$
such that

1. $X$ has a compatible linear ordering,
2. $(U_{s}(x)*V_{1}(x),…,U_{s}(x)*V_{m}(x),W_{1,s}(x),…,W_{n,s}(x))$
$\approx(V_{1}(x),…,V_{m}(x),y_{1,s},…,y_{n,s}),$
3. ind

4. there is some critical point $\alpha$ where $\text{crit}_{v}(y_{1,s},…,y_{n,s})=\alpha$ for all $s$, and
5. $y_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}y_{r,p}$.
6. $L(x,(y_{r,s})_{1\leq r\leq n,1\leq s\leq p})\neq 1$.
Challenge

I challenge the readers of this post to remove the large cardinal hypotheses from the above theorem (one may still use the freeness of subalgebras $\varprojlim_{n}A_{n}$ and the fact that $2*_{n}x=2^{n}\Rightarrow 1*_{n}x=2^{n}$ though).

So it turns out that by taking stronger large cardinal axioms, one can induce a linear ordering on the algebras of elementary embeddings without having to resort to working in forcing extensions. We say that a cardinal $\delta$ is an I1-tower cardinal if for all $A\subseteq V_{\delta}$ there is some $\kappa<\delta$ such that whenever $\gamma<\delta$ there is some cardinal $\lambda<\delta$ and non-trivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ such that $\text{crit}(j)=\kappa$ and where $j(\kappa)>\delta$ and where $j(A)=A$. If $A$ is a good enough linear ordering on $V_{\delta}$, then $A\cap V_{\lambda}$ induces a compatible linear ordering the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ such that $j(A\cap V_{\gamma})=A\cap V_{j(\gamma)}$ for all $\gamma<\lambda$. It is unclear where the I1-tower cardinals stand on the large cardinal hierarchy or whether they are even consistent.

It turns out that we can directly show that if $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ is a non-trivial elementary embedding, then there is a linear ordering $B$ of $V_{\lambda}$ where $j(B)=B$. In fact, if $j:V_{\lambda}\rightarrow V_{\lambda}$ is a non-trivial elementary embedding, $\mathrm{crit}(j)=\kappa$, and $A$ is a linear ordering of $V_{\lambda}$, then if we let $B=\bigcup_{n}j^{n}(A)$, then $B$ is a linear ordering of $V_{\lambda}$ and $j(B\cap V_{\gamma})=B\cap V_{j(\gamma)}$ whenever $\gamma<\lambda$. In particular, if $j$ extends to an elementary embedding $j^{+}:V_{\lambda+1}\rightarrow V_{\lambda+1}$, then $j^{+}(B)=B$. One can therefore prove the results about finite permutative LD-systems by working with the linear ordering that comes from $B$ instead of the linear ordering that comes from the fact that $V_{\lambda}[G]\models V=HOD$ in some forcing extension. One thing to be cautious of when one announces results before publication is that perhaps the proofs are not optimal and that one can get away with a simpler construction.

Philosophy and research project proposals

In the above results, we have worked in a model $V$ where there are non-trivial maps $j:V_{\lambda}\rightarrow V_{\lambda}$ and where $V_{\lambda}\models\text{V=HOD}$ in order to obtain compatible linear orderings on finite algebras. However, if we work in different forcing extensions with rank-into-rank embeddings instead, then I predict that one may obtain from large cardinals different results about finite algebras.

I predict that in the near future, mathematicians will produce many results about finite or countable self-distributive algebras using forcing and large cardinals where the large cardinal hypotheses cannot be removed. I also predict that rank-into-rank cardinals will soon prove results about structures that at first glance have little to do with self-distributivity.

The question of consistency

I must admit that I am not 100 percent convinced of the consistency of the large cardinals around the rank-into-rank level. My doubt is mainly due to the existence of finite reduced Laver-like LD-systems which cannot be subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$. However, if no inconsistency is found, then the results about finite or countable structures that arise from very large cardinals would convince me not only of the consistency of very large cardinals but also the existence of these very large cardinals. Therefore, people should investigate the finite algebras which arise from very large cardinals in order to quell all doubts about the consistency or the existence of these very large cardinals.

Since it is much more likely that the Reinhardt cardinals are inconsistent than say the I1 cardinals are inconsistent, I also propose that we attempt to use the algebras of elementary embeddings to show that Reinhardt cardinals are inconsistent. I have not seen anyone investigate the self-distributive algebras of elementary embeddings at the Reinhardt level. However, I think that investigating the self-distributive algebras of elementary embeddings would be our best hope in proving that the Reinhardt cardinals are inconsistent.

## Some new pages on Laver tables.

I have added the following new pages on the classical Laver tables.

The classical Laver tables can be given in ZFC a linear ordering such that the endomorphisms are monotone functions. When we reorder the classical Laver tables according to this compatible linear ordering we get the tables in the following link.
http://boolesrings.org/jvanname/lavertables-database-classical-fullcompatibletabletoa5/

The following link gives the compressed versions of the multiplication table for the classical Laver tables reordered according to the compatible linear ordering.
http://boolesrings.org/jvanname/lavertables-database-classical-alltablescompatibletoa10/

The compatible linear ordering on the classical Laver tables produces fractal like patterns that converge to a compact subset of the unit square. These images are found on the following link.
http://boolesrings.org/jvanname/lavertables-visualization-classical-imagesofcompatibletables/

And this page gives images of the fractal pattern that comes from the classical Laver tables.
http://boolesrings.org/jvanname/lavertables-visualization-classical-imagesoftables/

Hopefully I will be able to finish my long paper on the classical Laver tables over the next couple of months.

## Why complete regularity rather than Hausdorff is the correct cutoff point between lower and higher separation axioms

Disclaimer: By regular (completely regular) we mean regular (completely regular) and $T_{0}$

In general topology, there are two different kinds of topological spaces. There are the topological spaces that satisfy higher separation axioms such as the 3 dimensional space that we live in; when most people think of general topology (especially analysts and algebraic topologists), they usually think of spaces which satisfy higher separation axioms. On the other hand, there are topological spaces which only satisfy lower separation axioms; these spaces at first glance appear very strange since sequences can converge to multiple points. They feel much different from spaces which satisfy higher separation axioms. These spaces include the Zariski topology, finite non-discrete topologies, and the cofinite topology. Even spaces that set theorists consider such as the ordinal topology on a cardinal $\kappa$ or the Stone-Cech compactication $\beta\omega$ satisfy higher separation axioms; after all, $\beta\omega$ is the maximal ideal space of $\ell^{\infty}$. The general topology of lower separation axioms is a different field of mathematics than the general topology of higher separation axioms.

However, can we in good conscience formally draw the line between the lower separation axioms and the higher separation axioms or is the notion of a higher separation axiom simply an informal notion? If there is a line, then where do we draw the line between these two kinds of topological spaces?

As the sole owner of a silver badge in general topology on mathoverflow, I declare that the axiom complete regularity is the place where we need to draw the line between the lower separation axioms and the higher separation axioms. I can also argue that complete regularity is correct cutoff point by appealing to an authority greater than myself; the American Mathematical Society’s MSC-classification (the authority on classifying mathematics subjects) also delineates the lower separation axioms and the higher separation axioms at around complete regularity:
54D10-Lower separation axioms ($T_0$–$T_3$, etc.)
54D15-Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

Let me now give a few reasons why complete regularity is the pivotal separation axiom.

Hausdorffness is not enough. We need at least regularity.

Hausdorff spaces are appealing to mathematicians because Hausdorff spaces are precisely the spaces where each net (or filter) converges to at most one point. However, the condition that every net converges to at most one point should not be enough for a space to feel like it satisfies higher separation axioms. Not only do I usually want filters to converge to at most one point, but I also want the closures of the elements in a convergent filter to also converge. However, this condition is equivalent to regularity.

$\mathbf{Proposition}:$ Let $X$ be Hausdorff space. Then $X$ is regular if and only if whenever $\mathcal{F}$ is a filter that converges to a point $x$, the filterbase $\{\overline{R}\mid R\in\mathcal{F}\}$ also converges to the point $x$.

The next proposition formulates regularity in terms of the convergence of nets. The intuition behind the condition in the following proposition is that for spaces that satisfy higher separation axioms, if $(x_{d})_{d\in D},(y_{d})_{d\in D}$ are nets such that $x_{d}$ and $y_{d}$ get closer and closer together as $d\rightarrow\infty$, and if $(y_{d})_{d\in D}$ converges to a point $x$, then $(x_{d})_{d\in D}$ should also converge to the same point $x$.

$\mathbf{Proposition}$ Let $X$ be a Hausdorff space. Then $X$ is regular if and only if whenever $(x_{d})_{d\in D}$ is a net that does not converge to a point $x$, there are open neighborhoods $U_{d}$ of $x_{d}$ such that whenever $y_{d}\in U_{d}$ for $d\in D$, the net $(y_{d})_{d\in D}$ does not converge to the point $x$ either.

$\mathbf{Proof:}$ $\rightarrow$ Suppose that $(x_{d})_{d\in D}$ does not converge to $x$. Then there is an open neighborhood $U$ of $x$ where $\{d\in D\mid x_{d}\not\in U\}$ is cofinal in $D$. Therefore, there is some open set $V$ with $x\in V\subseteq\overline{V}\subseteq U$. Therefore, let $U_{d}=(\overline{V})^{c}$ whenever $d\in D$ and $U_{d}$ be an arbitrary set otherwise. Then whenever $y_{d}\in U_{d}$ for each $d\in D$, the set $\{d\in D\mid y_{d}\not\in U\}$ is cofinal in $D$. Therefore, $(y_{d})_{d\in D}$ does not converge to $x$ either.

$\leftarrow$ Suppose now that $X$ is not regular. Then there is an $x\in X$ and an open neighborhood $U$ of $x$ such that if $V$ is an open set with $x\in V$, then $V\not\subseteq U$. Therefore, let $D$ be a directed set and let $U_{d}$ be an open neighborhood of $x$ for each $d\in D$ such that for all open neighborhoods $V$ of $x$ there is a $d\in D$ so that if $e\geq d$, then $U_{d}\subseteq V$. Then let $x_{d}\in\overline{U_{d}}\setminus U$ for all $d\in D$. Then $(x_{d})_{d\in D}$ does not converge to $x$. Now suppose that $V_{d}$ is a neighborhood of $x_{d}$ for each $d\in D$. Then for each $d\in D$, we have $V_{d}\cap U_{d}\neq\emptyset$. Therefore, let $y_{d}\in V_{d}\cap U_{d}$. Then $(y_{d})_{d\in D}$ does converge to $x$. $\mathbf{QED}$.

Complete regularity is closed under most reasonable constructions

If there is a main separation axiom that draws the line between higher separation axioms and lower separation axioms, then this main separation axiom should be closed under constructions such as taking subspaces and taking arbitrary products. Since every completely regular space is isomorphic to a subspace $[0,1]^{I}$, the crossing point between lower and higher separation axioms should be no higher than complete regularity.

Not only are the completely regular spaces closed under taking products and subspaces, but the completely regular spaces are also closed under taking ultraproducts, the $P$-space coreflection, box products and other types of products, and various other constructions. Since we want our main separation axiom to be closed under most reasonable standard constructions and no lower than regularity, regularity and complete regularity are the only two candidates for our main separation axiom. We shall now find out why complete regularity is a better candidate than regularity for such a separation axiom.

Completely regular spaces can be endowed with richer structure

The completely regular spaces are precisely the spaces which can be given extra structure that one should expect to have in a topological space.

While a topological space gives one the notion of whether a point is touching a set, a proximity gives on the notion of whether two sets are touching each other. Every proximity space has an underlying topological space. Proximity spaces are defined in terms of points and sets with no mention of the real numbers, but proximity spaces are always completely regular. Furthermore, the compatible proximities on a completely regular space are in a one-to-one correspondence with the Hausdorff compactifications of the space.

$\mathbf{Theorem:}$ A topological space is completely regular if and only if it can be endowed with a compatible proximity.

The notion of a uniform space is a generalization of the notion of a metric space so that one can talk about concepts such as completeness, Cauchy nets, and uniform continuity in a more abstract setting. A uniform space gives one the notion of uniform continuity in the same way the a topological space gives one the notion of continuity. The definition of a uniform space is also very set theoretic, but it turns out that that every uniform space is induced by a set of pseudometrics and hence completely regular.

$\mathbf{Theorem:}$ A topological space is completely regular if and only if it can be endowed with a compatible uniformity.

For example, it is easy to show that every $T_{0}$-topological group can be given a compatible uniformity. Therefore, since the topological groups can always be given compatible uniformities, every topological group (and hence every topological vector space) is automatically completely regular.

Complete regularity is the proper line of demarcation between low and high separation axioms since the notions of a proximity and uniformity (which capture intuitive notions related to topological spaces without referring to the real numbers) induce precisely the completely regular spaces.

The Hausdorff separation axiom generalizes poorly to point-free topology

I realize that most of my readers probably have not yet been convinced of the deeper meaning behind point-free topology, but point-free topology gives additional reasons to prefer regularity or complete regularity over Hausdorffness.

Most concepts from general topology generalize to point-free topology seamlessly including separation axioms (regularity, complete regularity, normality), connectedness axioms (connectedness, zero-dimensionality, components), covering properties (paracompactness,compactness, local compactness, the Stone-Cech compactification), and many other properties. The fact that pretty much all concepts from general topology extend without a problem to point-free topology indicates that point-free topology is an interesting and deep subject. However, the notion of a Hausdorff space does not generalize very well from point-set topology to point-free topology. There have been a couple attempts to generalize the notion of a Hausdorff space to point-free topology. For example, John Isbell has defined an I-Hausdorff frame to be a frame $L$ such that the diagonal mapping $D:L\rightarrow L\oplus L$ is a closed localic mapping ($\oplus$ denotes the tensor product of frames). I-Hausdorff is a generalization of Hausdorffness since it generalizes the condition “$\{(x,x)\mid x\in X\}$ is closed” which is equivalent to Hausdorffness. Dowker and Strauss have also proposed several generalizations of Hausdorffness. You can read more about these point-free separation axioms at Karel Ha’s Bachelor’s thesis here. These many generalizations of the Hausdorff separation axioms are not equivalent. To make matters worse, I am not satisfied with any of these generalizations of Hausdorffness to point-free topology.

It is often the case that when an idea from general topology does not extend very well to point-free topology, then that idea relies fundamentally on points. For example, the axiom $T_{0}$ is completely irrelevant to point-free topology since the axiom $T_{0}$ is a pointed concept. Similarly, the axiom $T_{1}$ is not considered for point-free topology since the notion of a $T_{1}$-space is also fundamentally a pointed notion rather than a point-free notion. For a similar reason, Hausdorffness does not extend very well to point-free topology since the definition of Hausdorffness seems to fundamentally rely on points.

Just like in point-set topology, in point-free topology there is a major difference between the spaces which do not satisfy higher separation axioms and the spaces which do satisfy higher separation axioms. The boundary between lower separation axioms and higher separation axioms in point-set topology should therefore also extend to a boundary between lower separation axioms and higher separation axioms in point-free topology. Almost all the arguments for why complete regularity is the correct boundary between lower and higher separation axioms that I gave here also hold for point-free topology. Since Hausdorffness is not very well-defined in a point-free context, one should not regard Hausdorffness as the line of demarcation between lower separation axioms and higher separation axioms in either point-free topology or point-set topology.

Conclusion

Spaces that only satisfy lower separation axioms are good too.

While completely regular spaces feel much different from spaces which are not completely regular, spaces which satisfy only lower separation axioms are very nice in their own ways. For example, non $T_{1}$-spaces have a close connection with ordered sets since every non-$T_{1}$-space has a partial ordering known as the specialization ordering. I do not know much about algebraic geometry, but algebraic geometers will probably agree that spaces which only satisfy the lower separation axioms are important. Frames (point-free topological spaces) which only satisfy lower separation axioms are also very nice from a lattice theoretic point of view; after all, frames are precisely the complete Heyting algebras.

The underappreciation for complete regularity

The reason why Hausdorffness is often seen as a more important separation axiom than complete regularity is that Hausdorffness is easy to define than complete regularity. The definition of Hausdorffness only refers to points and sets while complete regularity refers to points, sets, and continuous real-valued functions. Unfortunately, since the definition of complete regularity is slightly more complicated than the other separation axioms, complete regularity is not often given the credit it deserves. For example, in the hierarchy of separation axioms, complete regularity is denoted as $T_{3.5}$. It is not even given an integer. However, Hausdorffness is denoted as $T_{2}$, regularity is denoted as $T_{3}$ and normality is denoted as $T_{4}$. Furthermore, when people often mention separation axioms they often fail to give complete regularity adequate attention. When discussing separation axioms in detail, one should always bring up and emphasize complete regularity.

In practice, the Hausdorff spaces that people naturally comes across are always completely regular. I challenge anyone to give me a Hausdorff space which occurs in nature or has interest outside of general topology which is not also completely regular. The only Hausdorff spaces which are not completely regular that I know of are counterexamples in general topology and nothing more. Since all Hausdorff spaces found in nature are completely regular, complete regularity should be given more consideration than it is currently given.

## Hello everyone

Hello everyone,

I am Joseph Van Name, and I have recently joined Booles’ Rings. I currently know several of the people here on Booles’ Rings through either mathoverflow.net or through the New York City logic community. I enjoy reading the mathematical posts here on Booles’ rings, and I am glad to be a part of this community.

I have requested to join Booles’ Rings in part due to my recent research endeavors towards understanding Laver tables. Through Booles’ Rings, I intend to post data, images, computer programs, and of course short mathematical expositions about these generalizations of the notion of a Laver table. Of course, I will also make posts about other areas of mathematics that I have researched in the past including publications and notes and slides for past talks. I therefore plan on having two portions of my site with one portion containing all the information on Laver tables one could ask for while the other portion is about all my other research projects.

Hopefully, through Booles’ Rings, I will use generalizations of Laver tables to establish a much needed common ground between set theory (in particular large cardinals) and structures such as self-distributive algebras, knots, braids and possibly other areas. By relating large cardinals to more conventional areas of mathematics, I intend to help non set-theorists see large cardinals not as being irrelevant objects that lie high above the clouds but as objects of a practical importance despite their astonishing size.