# Directed Sets of Topology – Tukey Representation and Rejection

Ziqing Feng and Paul Gartside

(August 2023)

###### Abstract

Every directed set is Tukey equivalent to (a) the family of all compact
subsets, ordered by inclusion, of a (locally compact) space, to (b) a
neighborhood filter, ordered by reverse inclusion, of a point (of a compact
space, and of a topological group), and to (c) the universal uniformity,
ordered by reverse inclusion, of a space. Two directed sets are Tukey
equivalent if they are cofinally equivalent in the sense that they can both be
order embedded _cofinally_ in a third directed set.

In contrast, any totally bounded uniformity is Tukey equivalent to
$[\kappa]^{<\omega}$, the collection of all finite subsets of $\kappa$, where
$\kappa$ is the cofinality of the uniformity. All other Tukey types are
‘rejected’ by totally bounded uniformities. Equivalently, a compact space $X$
has weight (minimal size of a base) equal to $\kappa$ if and only if the
neighborhood filter of the diagonal is Tukey equivalent to
$[\kappa]^{<\omega}$.

A number of questions from the literature are answered with the aid of the
above results.

Keywords: Directed set, Tukey order, compact set, neighborhood filter,
universal uniformity, totally bounded uniformity.

MSC Classification: 03E04, 06A07, 54D30, 54E15, 54E35, 54F05.

##  1 Introduction

Two directed sets are _Tukey equivalent_ if they are cofinally equivalent in
the sense that they can both be order embedded _cofinally_ in a third directed
set. Tukey equivalence [15] was originally introduced, early in the 20th
century, as a tool to understand convergence in general topological spaces,
however it was quickly seen to have broad applicability in comparing partial
orders. Key to the utility of Tukey equivalence (and more generally, Tukey
order) is that because it focuses on what happens cofinally it is sufficiently
coarse to allow comparison of very different directed sets, but nevertheless
preserves many order invariants. Fremlin [8] was the first to realize the
relevance of the Tukey order in analysis, showing that a fundamental result of
Bartoszynski and Raisonnier & Stern on additivity of the measure and category
ideals was due to the Tukey order relation between the relevant ideals.

Directed sets arise naturally in topology and in a variety of contexts. (In
this paper all topological spaces are Tychonoff.) Here we show that every
directed set can be represented, up to Tukey equivalence, by such a
topological directed set. Specifically, Theorem 2.2 states that for every
directed set $P$ there is a locally compact space $X_{P}$, a compact space
$K_{P}$ and point $x_{P}$, and a space $Y_{P}$ such that $P$ is Tukey
equivalent to $\mathcal{K}(X)$ (all compact subsets of $X$ ordered by
inclusion), $\mathcal{N}_{x_{P}}^{K_{P}}$ (all open neighborhoods of $x_{P}$
in $K_{P}$ ordered by reverse inclusion), and $\mathcal{U}_{Y_{P}}$ (the
universal uniformity of $Y_{P}$ ordered by reverse inclusion). In principle,
then, the study of arbitrary directed sets up to Tukey equivalence can be
restricted simply to that of $\mathcal{K}(X)$ (or neighborhood filters of
points in a compact space, or universal uniformities). Fortunately the Tukey
types of the directed set $\mathcal{K}(X)$ have been studied intensively [7,
10, 11, 12, 5].

Strikingly, in the opposite direction we show, see Theorem 3.1, that any
totally bounded uniformity is Tukey equivalent to $[\kappa]^{<\omega}$, the
collection of all finite subsets of $\kappa$, the cofinality of the uniformity
- all other Tukey types are ‘rejected’. A more topological interpretation of
this result, given in Theorem 3.5, is that a compact space $X$ has weight
(minimal size of a base) equal to $\kappa$ if and only if the neighborhood
filter of the diagonal is Tukey equivalent to $[\kappa]^{<\omega}$. Some
variations and open problems on this theme are discussed in Section 3.2.

A space $X$ is said to have a _$P$ -base_, where $P$ is a directed set, if
every point $x$ has a neighborhood base, $\\{U_{p}:p\in P\\}$, where
$U_{p}\subseteq U_{p^{\prime}}$ if $p\geq p^{\prime}$. It is easy to see that
a space with a compatible uniformity Tukey equivalent to some $P$ has a
$P$-base, and it is immediate that if a space has a $P$-base then it contains
a point whose neighborhood filter is Tukey equivalent to $P$. It follows from
the results outlined above that although in general (even in topological
groups) there are spaces with a $P$-base for every $P$ (up to Tukey
equivalence), for _compact_ spaces having a $P$-base for some $P$ is
delicately balanced between ‘everything goes’ and ‘everything rejected except
$[\kappa]^{<\omega}$’. This helps explain the recent surge of interest [1, 2,
3, 6] in spaces with $P$-bases, especially compacta, for certain nice $P$. One
particular outstanding question is whether compact, scattered spaces, of
countable scattered height with a $P$-base where $P$ has calibre
$(\omega_{1},\omega)$ are necessarily countable. Theorem 3.7 gives a positive
answer under a mild additional condition.

In the final section we apply the results above to answer a number (eleven) of
questions from the literature.

##  2 Representing Directed Sets

###  2.1 Directed Sets from Topology

For any space $X$, $\mathcal{K}(X)$ will denote the directed set of all
compact subsets of $X$, ordered by inclusion. If $A$ is a subset of $X$, then
$\mathcal{N}_{A}^{X}$ will denote the neighborhood filter of $A$ in $X$,
ordered by reverse inclusion. We abbreviate $\mathcal{N}_{\\{x\\}}^{X}$, the
neighborhood filter of a point $x$ in $X$, to $\mathcal{N}_{x}^{X}$. Along
with neighborhood filters of points, we pay particular attention to the
neighborhood filter, $\mathcal{N}_{\Delta}^{X^{2}}$, of the diagonal, and
certain subfilters, specifically compatible uniformities. Recall that a
(compatible) uniformity on $X$ is a subfilter, say $\mathcal{U}$, of
$\mathcal{N}_{\Delta}^{X^{2}}$ (such that for every $x$ in $X$ the family of
sets $U[x]=\\{y:(x,y)\in U\\}$, where $U$ is in $\mathcal{U}$, is a
neighborhood base at $x$ in $X$) where, for any $U$ we have that
$U^{-1}=\\{(y,x):(x,y)\in U\\}$ is in $\mathcal{U}$ and there is a $V$ in
$\mathcal{U}$ such that $V\circ V\subseteq U$, where $V\circ
V=\\{(x,z):(x,y),(y,z)\in V\\}$. The universal uniformity, $\mathcal{U}_{X}$,
is the finest compatible uniformity on $X$. We note that if $X$ is paracompact
then the universal uniformity and neighborhood filter of the diagonal
coincide. A uniformity, $\mathcal{U}$, on $X$ is totally bounded if for every
$U$ in $\mathcal{U}$ there is a finite subset $F$ of $X$ such that
$\\{U[x]:x\in F\\}$ covers $X$. Every space has at least one compatible
totally bounded uniformity. Every uniformity, $\mathcal{U}$, has a completion,
$\widehat{\mathcal{U}}$, defined on a superset, $\widehat{X}$, of $X$, and an
alternative description of totally bounded uniformities are those whose
completion is compact. (See [4] for a general reference on all topics
mentioned above.)

Let $P$ and $Q$ be directed sets. Then $Q$ is a _Tukey quotient_ of $P$,
denoted $P\geq_{T}Q$, if there is a _cofinal map_ $\phi$ from $P$ to $Q$, in
other words: for every cofinal subset $C$, say, of $P$ we have that $\phi(C)$
is cofinal in $Q$. Any map $\phi:P\to Q$ which is order-preserving and has
cofinal image is a cofinal map. A map $\psi:Q\to P$ is a called a _Tukey map_
if for every unbounded subset, $U$ say, of $Q$ we have that $\psi(U)$ is
unbounded in $P$. It turns out that there is a Tukey quotient from $P$ to $Q$
if and only if there is a Tukey map from $Q$ to $P$. Two directed sets $P$ and
$Q$ are _Tukey equivalent_ , denoted $P=_{T}Q$, if and only if $P\geq_{T}Q$
and $Q\geq_{T}P$. (This is equivalent to the definition of Tukey equivalence
stated in the introduction, namely $P$ and $Q$ embed cofinally in a third
directed set.)

######  Lemma 2.1.

Let $\mathcal{U}$ be a uniformity on $X$. Then the uniformity and its
completion are Tukey equivalent, $\mathcal{U}=_{T}\widehat{\mathcal{U}}$.

To see this, define $\phi:\mathcal{U}\to\widehat{\mathcal{U}}$ by
$\phi(U)=\overline{U}$ where the closure is taken in $\widehat{X}$, and
$\psi:\widehat{\mathcal{U}}\to\mathcal{U}$ by
$\psi(\widehat{U})=\widehat{U}\cap X^{2}$. Then $\phi$ and $\psi$ are order-
preserving and have cofinal image.

Note that in a topological group $G$, with identity element $e$, the left
translates (say) of neighborhoods of the identity gives a compatible
uniformity, $\mathcal{U}_{L}$, which is order isomorphic to
$\mathcal{N}_{e}^{G}$. Also, recall that in a compact space all compatible
uniformities are equal, and coincide with the family of all neighborhoods of
the diagonal. Now we show that _any_ directed set can be represented, up to
Tukey equivalence, by each of the types of directed sets arising in topology
introduced above.

######  Theorem 2.2.

Let $P$ be a directed set.

(1) $P=_{T}\mathcal{K}(X_{P})$ for some locally compact Hausdorff space
$X_{P}$.

(2) $P=_{T}\mathcal{N}_{x_{P}}^{K_{P}}$ for some compact Hausdorff space
$K_{P}$ and point $x_{P}$ in $K_{P}$.

(3) $P=_{T}\mathcal{N}_{e}^{G_{P}}$ for some topological group $G_{P}$ with
identity $e$.

(4) $P\times\omega=_{T}\mathcal{N}_{0}^{L_{P}}$ for some locally convex
topological vector space $L_{P}$.

(5) $P=_{T}\mathcal{U}_{Y_{P}}$, the fine uniformity, for some space $Y_{P}$.

(6) $P=_{T}\mathcal{N}_{\Delta}^{Y_{P}^{2}}$, for some space $Y_{P}$.

###### Proof.

First we establish (1). Let $D(P)$ be $P$ with the discrete topology. For a
$p$ in $P$ let $K_{p}=\overline{\,\downarrow\\!\\!p\,}$ where
$\,\downarrow\\!\\!p\,=\\{p^{\prime}\in P:p^{\prime}\leq p\\}$ and the closure
is in $\beta D(P)$. Note that $K_{p}$ is compact and open. Let
$X_{P}=\bigcup\\{K_{p}:p\in P\\}$ considered as a subspace of $\beta D(P)$.
Then $X_{P}$ is locally compact and Hausdorff. The map $\phi(p)=K_{p}$ is an
order-embedding of $P$ into $\mathcal{K}(X_{P})$ whose image is cofinal. To
see cofinality note that $\\{K_{p}:p\in P\\}$ is an open cover of $X_{P}$ so
any compact subset, $K$ say, is contained in finitely many
$K_{p_{1}},\ldots,K_{p_{n}}$, and now if $p$ is an upper bound of
$p_{1},\ldots,p_{n}$ then $K\subseteq\phi(p)$. Hence
$P=_{T}\mathcal{K}(X_{P})$.

Now for (2). By (1) we know there is a locally compact Hausdorff space $X_{P}$
such that $P$ and $\mathcal{K}(X_{P})$ are Tukey equivalent. Let $K_{P}$ be
the one-point compactification of $X_{P}$, and let $x_{P}$ be the point at
infinity. Then clearly $(\mathcal{K}(X_{P}),\subseteq)$ and
$(\mathcal{N}_{x_{P}}^{K_{P}},\supseteq)$ are order isomorphic, and so Tukey
equivalent.

For (3), let $X_{P}$ be the space in part (1), so $\mathcal{K}(X_{P})=_{T}P$,
and note it is zero dimensional. Set $G_{P}=C_{k}(X_{P},\mathbb{Z}_{2})$, the
group under co-ordinatewise addition modulo $2$ of all continuous maps of
$X_{P}$ into the discrete two point space, with the compact-open topology. The
identity of $G_{P}$ is $\mathbf{0}$, the constant zero function. Standard
basic open neighborhoods of $\mathbf{0}$ have the form $B(K)=\\{f\in
C(X,\mathbb{Z}_{2}):f(K)=0\\}$, for compact subsets $K$ of $X_{P}$. Clearly,
if $K\subseteq K^{\prime}$ then $B(K)\supseteq B(K^{\prime})$, and if $x\in
K\setminus K^{\prime}$ then the characteristic function of a clopen
neighborhood of $x$ disjoint from $K^{\prime}$ is in $B(K^{\prime})$ but not
$B(K)$. Thus $\phi:\mathcal{K}(X_{P})\to\mathcal{N}_{\mathbf{0}}^{G_{P}}$ is
an order isomorphism of $(\mathcal{K}(X_{P}),\subseteq)$ with a cofinal subset
of $(\mathcal{N}_{\mathbf{0}}^{G_{P}},\supseteq)$, and thus this latter
directed set is Tukey equivalent to $P$.

For (4), let $X_{P}$ be the space of part (1), and set $L_{P}=C_{k}(X_{P})$.
As shown in [13] $\mathcal{N}_{0}^{L_{P}}=_{T}\mathcal{K}(X_{P})\times\omega$,
and the claim follows.

Finally, we establish (5) and (6). We can suppose that $P$ is the neighborhood
filter, $\mathcal{N}_{x_{P}}^{X_{P}}$, of part (2). Rename the point $x_{P}$
to be $y_{P}$. Let $Y_{P}$ have underlying set $X_{P}$, and topology obtained
from $X_{P}$ by isolating all points except $y_{p}$. Then
$\mathcal{N}_{y_{p}}^{Y_{P}}=_{T}P$. Now $Y_{P}$ is paracompact, so
$\mathcal{U}_{Y_{P}}=\mathcal{N}_{\Delta}^{Y_{P}^{2}}$. The collection of all
$\Delta\cup U^{2}$ where $U$ is in $\mathcal{N}_{y_{P}}^{Y_{P}}$, is cofinal
in $\mathcal{N}_{y_{p}}^{Y_{P}}$ and order isomorphic to $P$. Claims (5) and
(6) follow. ∎

Regarding part (4) above we know (see [12]) which directed sets $Q$ are Tukey
equivalent to some $P\times\omega$.

######  Lemma 2.3.

Let $Q$ be a directed set. Then the following are equivalent: (i)
$Q=_{T}P\times\omega$ for some directed set $P$, (ii) $Q\geq_{T}\omega$, and
(iii) $Q$ is not countably directed.

###  2.2 Cofinality, Calibres, and Other Properties of $\mathcal{K}(X)$

The _cofinality_ of a directed set $P$, denoted $\operatorname{cof}(P)$, is
the minimal size of a cofinal subset of $P$. If $P\geq_{T}Q$ then
$\operatorname{cof}(P)\geq\operatorname{cof}(Q)$. Indeed,
$\operatorname{cof}(P)\leq\kappa$ if and only if
$[\kappa]^{<\omega}\geq_{T}P$. A directed set $P$ has _calibre
$(\mu,\lambda)$_ where $\mu$ and $\lambda$ are cardinals, $\mu$ regular, if
every $\mu$-sized subset of $P$ contains a $\lambda$-sized subset which is
bounded. If $P\geq_{T}Q$ and $P$ has calibre $(\mu,\lambda)$ then so does $Q$.
Indeed, $P\not\geq_{T}[\mu]^{<\omega}$ if and only if $P$ has calibre
$(\mu,\omega)$.

A _topological directed set_ is a directed set with a topology. For example,
if $X$ is any space, then $\mathcal{K}(X)$ is directed as usual by inclusion,
and is naturally equipped with the Vietoris topology. A topological directed
set $P$ is said to be _KSB_ (compact sets bounded) if every compact subset of
$P$ is bounded (above), and _DK_ (down sets compact) if every down set,
$\,\downarrow\\!\\!p\,=\\{p^{\prime}\in P:p^{\prime}\leq p\\}$, of $P$ is
compact. Observe that $\mathcal{K}(X)$ is KSB and DK.

######  Lemma 2.4.

Let $Q$ be a topological directed set. If $Q$ is locally compact,
$e(Q)\leq\kappa$, and $Q$ is KSB, then $Q$ has calibre $(\kappa^{+},\omega)$.

###### Proof.

Let $S$ be a subset of $Q$ of size $\kappa^{+}$. We show $S$ has an infinite
subset with an upper bound. As $e(Q)\leq\kappa$, $S$ is not closed and
discrete, so there is a $q$ in $Q$ such that $q$ is in the closure of
$S\setminus\\{q\\}$. Since $Q$ is locally compact there is a compact
neighborhood $C$ of $q$. Then $C$ contains an infinite subset $S_{0}$ of $S$.
Then $\overline{S_{0}}$ is compact, so by KSB, $S_{0}$ has an upper bound. ∎

A space $X$ is said to have a $P$-ordered compact cover, where $P$ is a
directed set, if it has a compact cover $\\{K_{p}:p\in P\\}$ where $p\leq
p^{\prime}$ implies $K_{p}\subseteq K_{p^{\prime}}$. Clearly if
$P\geq_{T}\mathcal{K}(X)$ then $X$ has a $P$-ordered compact cover. Let us say
that a space $X$ has _relative calibre $(\mu,\omega)$_ (in $\mathcal{K}(X)$)
if every subset $S$ of $X$ of size $\mu$ has an infinite subset $S_{0}$ with
compact closure. Clearly, if $\mathcal{K}(X)$ has calibre $(\mu,\omega)$ then
$X$ has relative calibre $(\mu,\omega)$. But also easy to check:

######  Lemma 2.5.

If a space $X$ has a $P$-ordered compact cover, where $P$ has calibre
$(\mu,\omega)$ then $X$ is relative calibre $(\mu,\omega)$.

Define the _extent_ of a space $Y$, to be $e(Y)=\sup\\{|E|:E$ is closed and
discete in $Y\\}$. Observe $Y$ has countable extent if and only if
$e(Y)\leq\aleph_{0}$.

######  Lemma 2.6.

If $X$ has relative calibre $(\kappa^{+},\omega)$ then $e(X)\leq\kappa$.

###### Proof.

To show that $e(X)\leq\kappa$ we need to show that no subset of $X$ of size
$>\kappa$ is closed discrete. Well take, any $S$ a subset of $X$ of size
$>\kappa$. Then by relative calibre $(\kappa^{+},\omega)$, there is an
infinite subset $S_{0}$ of $S$ with compact closure. Then $S_{0}$, and $S$,
can not be closed and discrete. ∎

######  Lemma 2.7.

For any space $X$ we have $\mathcal{K}(X)=_{T}\mathcal{K}(\mathcal{K}(X))$.

###### Proof.

Since $\mathcal{K}(X)$ embeds as a closed subspace of
$\mathcal{K}(\mathcal{K}(X))$, certainly
$\mathcal{K}(X)\leq_{T}\mathcal{K}(\mathcal{K}(X))$. For the converse, define
$\phi:\mathcal{K}(X)\to\mathcal{K}(\mathcal{K}(X))$ by
$\phi(K)=\,\downarrow\\!\\!K\,=\\{L\in\mathcal{K}(X):L\subseteq K\\}$.
Recalling that $\mathcal{K}(\mathcal{K}(X))$ is DK, we see $\phi$ does map
into $\mathcal{K}(\mathcal{K}(X))$. Clearly $\phi$ is order-preserving. For
any $\mathcal{K}$ in $\mathcal{K}(\mathcal{K}(X))$, we know
$\bigcup\mathcal{K}$ is a compact subset of $X$, then
$\phi(\bigcup\mathcal{K})\supseteq\mathcal{K}$, and so $\phi$ has cofinal
image. ∎

######  Lemma 2.8.

Let $X$ be locally compact. Then $\mathcal{K}(X)$ has calibre
$(\kappa^{+},\omega)$ if and only if $e(\mathcal{K}(X))\leq\kappa$.

###### Proof.

Suppose, first, that $\mathcal{K}(X)$ has calibre $(\kappa^{+},\omega)$. Then
$\mathcal{K}(\mathcal{K}(X))$ has calibre $(\kappa^{+},\omega)$, and
$\mathcal{K}(X)$ has relative calibre $(\kappa^{+},\omega)$ in
$\mathcal{K}(\mathcal{K}(X))$. Hence $e(\mathcal{K}(X))\leq\kappa$.

Now suppose, $e(\mathcal{K}(X))\leq\kappa$. Apply Lemma 2.4. ∎

##  3 ‘Rejecting’ Directed Sets in the Compact Case

###  3.1 Totally Bounded Uniformities

######  Theorem 3.1.

Let $X$ be a set. For any totally bounded uniformity, $\mathcal{U}$, on $X$ we
have $\mathcal{U}=_{T}[\kappa]^{<\omega}$ where
$\kappa=\operatorname{cof}(\mathcal{U})$. More generally, for any uniformity
$\mathcal{U}$ on $X$ and totally bounded subset $S$ we have
$\mathcal{U}\restriction S=_{T}[\kappa]^{<\omega}$ where
$\kappa=\operatorname{cof}(\mathcal{U}\restriction S)$.

###### Proof.

Evidently, the ‘more generally’ claim follows from the first part, which we
now prove.

For any directed set $P$ we have $[\operatorname{cof}(P)]^{<\omega}\geq_{T}P$,
so certainly $[\kappa]^{<\omega}\geq_{T}\mathcal{U}$. We show
$[\kappa]^{<\omega}\leq_{T}\mathcal{U}$. To do this it suffices to show that
there is a $\kappa$-sized subcollection, $\\{U_{\alpha}:\alpha<\kappa\\}$, of
$\mathcal{U}$ such that no infinite subset has an upper bound, for then
$\psi:[\kappa]^{<\omega}\to\mathcal{U}$ defined by $\psi(F)=\bigcap_{\alpha\in
F}U_{\alpha}$ is a Tukey map (carries unbounded sets to unbounded sets).

We may assume $X$ is compact in the topology induced by $\mathcal{U}$, and so
$\mathcal{U}=\mathcal{N}_{\Delta}^{X^{2}}$. Indeed we can replace $X$ with the
space completion, $\widehat{X}$, which is compact as $\mathcal{U}$ is totally
bounded, and $\mathcal{U}$ with its completion $\widehat{\mathcal{U}}$, which
is totally bounded. To see this, recall, see Lemma 2.1, that
$\mathcal{U}=_{T}\widehat{\mathcal{U}}$, so, in particular,
$\kappa=\operatorname{cof}(\mathcal{U})=\operatorname{cof}(\widehat{\mathcal{U}})$.

Construct by recursion a family of pairs $(x_{\alpha,1},x_{\alpha,2})$ in
$X^{2}$ and $U_{\alpha}$ in $\mathcal{U}$, for $\alpha<\kappa$, such that
$(\bigstar)$: $x_{\alpha,i}\neq x_{\beta,j}$ if $i\neq j$ or $\alpha\neq\beta$
and, if $x_{\beta,i}\in U_{\alpha}[x_{\alpha,i}]$ and $x_{\alpha,i}\in
U_{\beta}[x_{\beta,i}]$ for $i=1,2$ then $\alpha=\beta$. To facilitate the
construction we additionally ensure that for every $\alpha$ we have
$\overline{U_{\alpha}[x_{\alpha,1}]}\cap\overline{U_{\alpha}[x_{\alpha,2}]}=\emptyset$.

At stage $\alpha$ of the recursive construction first observe that
$\\{U_{\beta}[x_{\beta,1}]\times U_{\beta}[x_{\beta,2}]:\beta<\alpha\\}$ is
not a cover of $X^{2}\setminus\Delta$. Otherwise every compact (equivalently,
closed) subset of $X^{2}\setminus\Delta$ is contained in some finite union of
these open rectangles, and so the collection of all complements in $X^{2}$ of
finite unions of closures of rectangles $U_{\beta}[x_{\beta,1}]\times
U_{\beta}[x_{\beta,2}]$, where $\beta<\alpha$, would be a base for
$\mathcal{N}_{\Delta}^{X^{2}}=\mathcal{U}$ of size strictly less than
$\kappa$, which is the cofinality of $\mathcal{U}$ – contradiction. Pick
$(x_{\alpha,1},x_{\alpha,2})$ in $X^{2}\setminus\Delta$ but not in
$\bigcup_{\beta<\alpha}(U_{\beta}[x_{\beta,1}]\times U_{\beta}[x_{\beta,2}])$.
Pick $U_{\alpha}$ in $\mathcal{U}$ so that
$\overline{U_{\alpha}[x_{\alpha,1}]}\cap\overline{U_{\alpha}[x_{\alpha,2}]}=\emptyset$.
By construction, $x_{\alpha,1}\neq x_{\alpha,2}$, and for no $\beta<\alpha$ do
we have $x_{\alpha,i}\in U_{\beta}[x_{\beta,i}]$ for $i=1,2$. Hence
$(\bigstar)$ holds.

To complete the proof it remains to show that $\\{U_{\alpha}:\alpha<\kappa\\}$
contains no infinite bounded subcollection. For a contradiction, suppose $A$
is an infinite subset of $\kappa$ such that $U_{A}=\bigcap_{\alpha\in
A}U_{\alpha}$ is in $\mathcal{U}$. Pick $U_{\infty}$ in the uniformity
$\mathcal{U}$ so that $U_{\infty}^{-1}\circ U_{\infty}\subseteq U_{A}$. As
$\mathcal{U}$ is totally bounded, fix finite $F$ such that $U_{\infty}[F]=X$.
Then $\\{U_{\infty}[a]\times U_{\infty}[b]:a,b\in F\\}$ is a finite cover of
$X^{2}$, so there is an infinite subset $A^{\prime}$ of $A$ and
$(x_{\infty,1},x_{\infty,2})$ so that for all $\alpha$ in $A^{\prime}$ we have
$(x_{\alpha,1},x_{\alpha,2})\in U_{\infty}[x_{\infty,1}]\times
U_{\infty}[x_{\infty,2}]$.

Take any two distinct $\alpha,\beta$ from $A^{\prime}$. Then, for $i=1,2$, we
have $x_{\alpha,i}\in U_{\infty}[x_{\infty,i}]$ so (a)
$(x_{\alpha,i},x_{\infty,i})\in U_{\infty}^{-1}$, and $x_{\beta,i}\in
U_{\infty}[x_{\infty,i}]$ so (b) $(x_{\infty,i},x_{\beta,i})\in U_{\infty}$.
From (a) and (b), for $i=1,2$, we have $(x_{\alpha,i},x_{\beta,i})\in
U_{\infty}^{-1}\circ U_{\infty}\subseteq U_{\alpha}$, so $x_{\beta,i}$ is in
$U_{\alpha}[x_{\alpha,i}]$. Interchanging $\alpha$ and $\beta$ in the argument
above, for $i=1,2$, along with $x_{\beta,i}\in U_{\alpha}[x_{\alpha,i}]$, we
also have $x_{\alpha,i}\in U_{\beta}[x_{\beta,i}]$, and this contradicts
$(\bigstar)$, as desired. ∎

We now re-interpret this theorem, which is a combinatorial statement about the
Tukey type of certain uniformities (namely the totally bounded ones)
topologically and in terms of calibres. Recalling that
$P\not\geq_{T}[\mu]^{<\omega}$ if and only if $P$ has calibre $(\mu,\omega)$,
and applying this to $\mu=\kappa^{+}$, the following is immediate.

######  Theorem 3.2.

Suppose $\mathcal{U}$ is a compatible uniformity for a space $X$. If $A$ is a
totally bounded subset and $\mathcal{U}$ is calibre $(\kappa^{+},\omega)$ then
$w(A)\leq\kappa$.

Totally bounded subsets of uniform spaces are also called _precompact_ subsets
(because they are precisely those subsets whose closure in the completion is
compact). Pseudocompact, and, a fortiori, countably compact and compact
subsets are totally bounded with respect to any compatible uniformity.

######  Corollary 3.3.

Suppose $\mathcal{U}$ is a compatible uniformity for a space $X$. If $A$ is a
pseudocompact subspace and $\mathcal{U}$ is calibre $(\kappa^{+},\omega)$ then
$w(A)\leq\kappa$.

In particular, if $\mathcal{U}$ is calibre $(\omega_{1},\omega)$ then every
pseudocompact (and, countably compact or compact) subspace of $X$ is (compact,
second countable and) metrizable.

######  Corollary 3.4.

Let $G$ be a topological group with identity $e$. If $A$ is a precompact
subset of $G$, and $\mathcal{N}_{e}^{G}$ is calibre $(\kappa^{+},\omega)$ then
$w(A)\leq\kappa$.

Compact spaces have a unique compatible uniformity, which coincides with all
neighborhoods of the diagonal, and so with complements of compact subsets of
the square disjoint from the diagonal. Consequently we can rephrase again for
compact spaces, or subspaces, without (direct) reference to a uniformity.
Further recall (Lemma 2.8), if $Y$ is locally compact then $\mathcal{K}(Y)$
has calibre $(\kappa^{+},\omega)$ if and only if the extent of
$\mathcal{K}(Y)$, $e(\mathcal{K}(Y))$, is no more than $\kappa$.

######  Theorem 3.5.

Let $X$ be a space.

(1) Suppose $X$ is compact. Then $w(X)=\kappa$ if and only if
$\mathcal{N}_{\Delta}^{X^{2}}=_{T}[\kappa]^{<\omega}$.

Alternatively phrased, $w(X)\leq\kappa$ if and only if
$\mathcal{N}_{\Delta}^{X^{2}}$ has calibre $(\kappa^{+},\omega)$. And a second
alternative phrasing, $w(X)=e(\mathcal{K}(X^{2}\setminus\Delta))$.

(2) More generally, if $A$ is a compact subset of a space $X$, and
$\mathcal{N}_{\Delta}^{X^{2}}$ is calibre $(\kappa^{+},\omega)$ then
$w(A)\leq\kappa$.

Both (1) and (2) hold for any directed set Tukey equivalent to
$\mathcal{N}_{\Delta}^{X^{2}}$, including $\mathcal{U}_{X}$, or any other
compatible uniformity, and $\mathcal{K}(X^{2}\setminus\Delta)$.

###  3.2 Variations and Problems

Comparing what we can prove about pseudocompact and countably compact subsets
of uniform spaces with calibre $(\omega_{1},\omega)$, with what we can show
for compact subsets of a space, $X$, with $\mathcal{N}_{\Delta}^{X^{2}}$ or
$\mathcal{K}(X^{2}\setminus\Delta)$ having calibre $(\omega_{1},\omega)$, the
following questions naturally arise.

######  Question 1.

Let $X$ be a space. Suppose one of the directed sets specified has calibre
$(\omega_{1},\omega)$: (A) $\mathcal{N}_{\Delta}^{X^{2}}$ or (B)
$\mathcal{K}(X^{2}\setminus\Delta)$.

(i) If $X$ is countably compact then is $X$ (compact, second countable and)
metrizable?

(ii) If $X$ is pseudocompact then is $X$ (compact, second countable and)
metrizable?

(iii) If $A$ is a countably compact (closed) subset of $X$ then is $A$
(compact, second countable and) metrizable?

(iv) If $A$ is a pseudocompact (closed) subset of $X$ then is $A$ (compact,
second countable and) metrizable?

Next we answer (A)(i) positively. The remaining questions are wide open.

######  Proposition 3.6.

Let $X$ be a space such that $\mathcal{N}_{\Delta}^{X^{2}}$ has calibre
$(\omega_{1},\omega)$. If $X$ is countably compact then $X$ is compact (and
metrizable).

###### Proof.

We show the contra-positive. So suppose $X$ is not compact. Then either $X$ is
not countably compact, and we are done, or $X$ is not Lindelöf, so we can find
a strictly increasing sequence of open sets, $\\{U_{\alpha}:\alpha<\kappa\\}$
covering $X$ with no countable subcover (so $\kappa$ is uncountable). Fix
$y_{\alpha}\in U_{\alpha}\setminus\bigcup_{\beta<\alpha}U_{\beta}$. For each
$\alpha<\kappa$, let $V_{\alpha}=X\setminus\\{y_{\alpha}\\}$ and
$N_{\alpha}=U_{\alpha}^{2}\cup V_{\alpha}^{2}$ , which is a neighborhood of
the diagonal. As $\mathcal{N}_{\Delta}^{X^{2}}$ has calibre
$(\omega_{1},\omega)$, we know there is an infinite $A\subseteq\kappa$ and a
neighborhood $N$ of the diagonal such that $N_{\alpha}\supseteq N$ for all
$\alpha\in A$. Without loss of generality, we may assume
$N=\bigcup_{W\in\mathcal{W}}W^{2}$ for some open cover $\mathcal{W}$ of $X$.
We will show that each member of $\mathcal{W}$ contains at most one point of
the infinite set $\\{y_{\alpha}:\alpha\in A\\}$. Indeed, if
$y_{\alpha},y_{\beta}\in W$ for some $W$ from $\mathcal{W}$ and distinct
$\alpha,\beta\in A$, then we see that $(y_{\alpha},y_{\beta})\in
W^{2}\subseteq N\subseteq N_{\alpha}=U_{\alpha}^{2}\cup V_{\alpha}^{2}$ . Now
since $y_{\alpha}\notin V_{\alpha}$ , then we must have $y_{\beta}\in
U_{\alpha}$ , which gives $\beta\leq\alpha$. But symmetrically,
$(y_{\alpha},y_{\beta})\in U_{\beta}^{2}\cup V_{\beta}^{2}$ , which implies
that $\alpha\leq\alpha$. Thus $\alpha=\beta$. Hence, $\mathcal{W}$ witnesses
that $\\{y_{\alpha}:\alpha\in A\\}$ is an infinite closed discrete subset of
$X$, and thus $X$ is not countably compact. ∎

###  3.3 $P$-Bases

Recall from the Introduction that a space $X$ is said to have a $P$-base,
where $P$ is a directed set, if every point $x$ has a neighborhood base,
$\\{U_{p}:p\in P\\}$, where $U_{p}\subseteq U_{p^{\prime}}$ if $p\geq
p^{\prime}$. Also recall that a topological space $X$ is scattered if each
non-empty subspace of $X$ has an isolated point. It is known that any compact
scattered space is zero-dimensional. Scattered spaces can be stratified by the
scattered height, as follows. For any subspace $A$ of a space $X$, let
$A^{\prime}$ be the set of all non-isolated points of $A$. It is
straightforward to see that $A^{\prime}$ is a closed subset of $A$. Let
$X^{(0)}=X$ and define
$X^{(\alpha)}=\bigcap_{\beta<\alpha}(X^{(\beta)})^{\prime}$ for each
$\alpha>0$. Then a space $X$ is scattered if $X^{(\alpha)}=\emptyset$ for some
ordinal $\alpha$. If $X$ is scattered then for each of its points, $x$, there
exists a unique ordinal $h(x)$ such that $x\in X^{(h(x))}\setminus
X^{(h(x)+1)}$. The ordinal $h(X)=\sup\\{h(x):x\in X\\}$ is called the
scattered height of $X$ and is denoted by $h(X)$. Also, it straightforward to
show that for any compact scattered space $X$, $X^{(h(X))}$ is a non-empty
finite subset.

The ordinal space $\omega_{1}+1$ is compact, scattered, of scattered height
$\omega_{1}$, with a $P$-base for $P=\mathcal{K}(\mathbb{Q})$, (and
consistently, $P=\omega^{\omega}$), and so for $P$ with calibre
$(\omega_{1},\omega)$. But it is an interesting open question, raised in [6,
Question 3.3] , whether every compact scattered space, with countable
scattered height and a $P$-base where $P$ is calibre $(\omega_{1},\omega)$, is
countable. We show the answer is positive if the space is additionally
hereditarily meta-Lindelöf (every open cover of any subspace has a point-
countable open refinement).

######  Theorem 3.7.

Let $X$ be a hereditarily meta-Lindelöf, compact, scattered space with
countable scattered height. If $X$ has a $P$-base with $P$ having calibre
$(\omega_{1},\omega)$, then $X$ is countable, hence metrizable.

This follows immediately from the next technical result.

######  Theorem 3.8.

Let $X$ be a compact scattered space with countable scattered height. Suppose
that, to each $x$ in $X$, we can assign a clopen neighborhood $U_{x}$ such
that $U_{x}\cap X^{(h(x))}=\\{x\\}$ and $\\{U_{x}:x\in X\\}$ is point-
countable. If $X$ has a $P$-base with $P$ having calibre
$(\omega_{1},\omega)$, then $X$ is countable, hence metrizable.

###### Proof.

The proof is by induction on the scattered height. In [3] it is shown that the
result holds when the scattered height is finite. Assume, then, that the
scattered height of $X$ is $\alpha$ with $\alpha<\omega_{1}$ and the result
holds for any compact space with scattered height $<\alpha$. We show that $X$
is countable.

Suppose, first, $\alpha$ is a successor , $\alpha=\alpha^{-}+1$. Let
$Y=X^{(\geq\alpha^{-})}$. Clearly $Y$ is a compact subspace of $X$ with
scattered height $2$, hence it is countable. Then it is straightforward to
verify that $X$ is countable.

Now suppose that $\alpha$ is a limit. Without loss of generality, we assume
that $X^{(\alpha)}$ is a singleton, denoted by $x_{\ast}$. As $X$ has a
$P$-base we can fix a clopen base $\\{B_{p}:p\in P\\}$ at $x_{\ast}$, where
$B_{p}\subseteq B_{p^{\prime}}$ if $p\geq p^{\prime}$. For each $p\in P$,
$K_{p}=X\setminus B_{p}$ is compact and with scattered height $\beta_{p}$
which is clearly $<\alpha$, hence it is countable and also
$K_{p}^{(\beta_{p})}$ is finite. Note that $\\{K_{p}:p\in P\\}$ is a
$P$-ordered compact cover of $X\setminus\\{x_{\ast}\\}$.

Assume, for a contradiction, that $X$ is uncountable. Let $\\{U_{x}:x\in X\\}$
be the collection of clopen sets satisfying the conditions in the statement of
this theorem. Clearly for each $x\in X\setminus\\{x_{\ast}\\}$, $U_{x}$ is
compact with scattered height $<\alpha$, hence is countable. Also, without
loss of generality, we assume that $X\setminus X^{(1)}$ is uncountable. For
each $y\in X\setminus X^{(1)}$, let $C_{y}=\\{x\in
X\setminus\\{x_{\ast}\\}:y\in U_{x}\\}$ which is clearly countable and
$D_{y}=\bigcup\\{U_{x}:x\in C_{y}\\}$ which is countable too since each
$U_{x}$ for $x\in C_{y}$ is countable. By a transfinite induction, we pick a
set $\\{y_{\lambda}:\lambda<\omega_{1}\\}\subseteq X\setminus X^{(1)}$ such
that $y_{\lambda}\notin\bigcup\\{D_{y_{\gamma}}:\gamma<\lambda\\}$. Such a set
exists because $X\setminus X^{(1)}$ is uncountable. Also, it is clear that the
cardinality of $U_{x}\cap\\{y_{\lambda}:\lambda<\omega_{1}\\}$ is at most $1$.
For each $\lambda<\omega_{1}$, pick $p_{\lambda}\in P$ such that
$y_{\lambda}\in K_{p_{\lambda}}$. Since $P$ has calibre $(\omega_{1},\omega)$,
there is a countable subset $\\{\lambda_{n}:n\in\omega\\}$ in $\omega_{1}$
such that $\\{p_{\lambda_{n}}:n\in\omega\\}$ is bounded above. We denote
$p_{\ast}$ to be an upper bound of $\\{p_{\lambda_{n}}:n\in\omega\\}$ in $P$.
Then $\\{y_{\lambda_{n}}:n\in\omega\\}\subseteq K_{p_{\ast}}$. Since
$K_{p_{\ast}}$ is compact, there is a finite subset $\\{x_{i}:i\leq m\\}$ of
$K_{p_{\ast}}$ such that $K_{p_{\ast}}\subseteq\bigcup\\{U_{x_{i}}:i\leq
m\\}$. Then, $\\{y_{\lambda_{n}}:n\in\omega\\}\subseteq\\{U_{x_{i}}:i\leq
m\\}$ which is a contradiction because each $y_{\lambda_{n}}$ is at most in
one of the $\\{U_{x_{i}}:i\leq m\\}$. This finishes the proof. ∎

##  4 Examples and Applications

###  4.1 Strong Diagonals

We present as an application of Theorem 3.5 the following partial solution to
[14, Problem 4.1]. Partial because Sanchez asked about compact spaces, $X$,
with an ‘$M$-diagonal’ which means that $X^{2}\setminus\Delta$ has a
$\mathcal{K}(M)$-ordered compact cover, and we require a ‘strong
$M$-diagonal’. A space $X$ has a _strong $M$-diagonal_ if
$\mathcal{K}(M)\geq_{T}\mathcal{K}(X^{2}\setminus\Delta)$.

######  Theorem 4.1.

Suppose $M$ is a metric space and $K$ is a compact space with a strong
$M$-diagonal. Then $w(K)\leq w(M)$.

###### Proof.

First note that because every convergent sequence (say, $(K_{n})_{n}$
converging to $K_{\infty}$) in $\mathcal{K}(M)$ is bounded (by
$K_{\infty}\cup\bigcup_{n}K_{n}$) and every subset of $\mathcal{K}(M)$ of size
strictly bigger than $w(M)$ has a proper limit point, we have that
$\mathcal{K}(M)$ is calibre $(w(M)^{+},\omega)$. Then, because
$\mathcal{K}(M)\geq_{T}\mathcal{K}(X^{2}\setminus\Delta)$ we see
$\mathcal{N}_{\Delta}^{X^{2}}$ is calibre $(w(M)^{+},\omega)$. Now apply
Theorem 3.5. ∎

Sanchez’s [14, Problem 4.2] remains an interesting open problem, however the
next example gives a counter-example, at least consistently, to his remaining
Problems 4.3-4.10.

######  Example 4.2.

There are spaces $X$ and $Y$ with
$\mathcal{K}(Y)\geq_{T}\mathcal{K}(X^{2}\setminus\Delta)$ (i.e. ‘$X$ has a
strong $Y$-diagonal’) such that:

(1) $X$ is a first countable compact space, with $w(X)=\mathfrak{c}$, and $Y$
is $\sigma$-compact, first countable and cosmic;

(2) assuming $(\neg CH)$, taking $\kappa=\aleph_{1}$, $X$ is first countable
and compact, $Y$ is the union of $\kappa$-many compact sets, and
$nw(Y)\leq\kappa<w(X)$.

###### Proof.

Let $Y$ be the bowtie space. Let $X$ be any first countable, compact space of
weight $\mathfrak{c}$ (the double arrow space, for example, or the Alexandrov
duplicate of $[0,1]$). Then $X$ is compact, first countable and, assuming
$(\neg CH)$, $w(X)=\mathfrak{c}>\aleph_{1}=\kappa$. While $Y$ is
$\sigma$-compact (hence is the union of $\aleph_{1}$-many compact subsets),
$nw(Y)=\aleph_{0}\leq\aleph_{1}$. Since $X$ has weight $\mathfrak{c}$, by
Theorem 3.5 we know
$\mathcal{K}(X^{2}\setminus\Delta)=_{T}[\mathfrak{c}]^{<\omega}$. But from [3]
we know, as $Y$ is the bowtie space,
$\mathcal{K}(Y)=_{T}[\mathfrak{c}]^{<\omega}$ Hence,
$\mathcal{K}(Y)\geq_{T}\mathcal{K}(X^{2}\setminus\Delta)$, as claimed. ∎

###  4.2 General Topological Groups

Solving, negatively, Question 6.5 from [9] and Question 5.9 from [6] we offer
the following example.

######  Example 4.3.

There is an Abelian topological group $G$, with identity $0$, such that all
precompact subsets of $G$ are metrizable and
$\chi(G)=\operatorname{cof}(\mathcal{N}_{0}^{G})=\mathfrak{d}$ but for no
separable metrizable space $M$ do we have
$\mathcal{K}(M)\geq_{T}\mathcal{N}_{0}^{G}$, in particular,
$\omega^{\omega}\not\geq_{T}\mathcal{N}_{0}^{G}$ ($G$ does not have an
$\omega^{\omega}$-base).

###### Proof.

Let $P=\sum\omega^{\omega_{1}}$ (the $\Sigma$ product of $\omega_{1}$-many
copies of $\omega$). Then $P$ is calibre $(\omega_{1},\omega)$ [11]. And the
cofinality of $P$ is $\mathfrak{d}$ (for each $\alpha<\omega_{1}$ take a
cofinal family, $C_{\alpha}$, of size $\leq\mathfrak{d}$ of $\alpha^{\omega}$,
extend each element of $C_{\alpha}$ to have value $0$ for all
$\beta\geq\alpha$, giving a $C_{\alpha}^{\prime}$ contained in
$\sum\omega^{\omega_{1}}$, and union together the $C_{\alpha}^{\prime}$’s).
For no separable metrizable $M$ [11] do we have
$\mathcal{K}(M)\geq_{T}\sum\omega^{\omega_{1}}$, in particular not
$\omega^{\omega}$.

Now construct the topological group, $G=G_{P}$, as in Theorem 2.2(3).
Corollary 3.4 guarantees that precompact subsets of $G$ are metrizable. ∎

## References

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|  $\begin{aligned} &V^{\dagger}_{ik}\left[[{{C}}_{euLL}^{V}]^{\alpha\beta kl}-\chi\,\left(k_{e_{L}}\,[{{\mathbf{c}}}_{u_{L}Z}]^{kl}\,\delta_{\alpha\beta}+k_{u_{L}}\,[{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{kl}\right)\right]V_{{\ell}j}\\\ &~{}=U^{\dagger}_{\alpha\rho}\left[[{{C}}_{\nu dLL}^{V}]^{\rho\sigma ij}-\left(k_{\nu_{L}}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\rho\sigma}+k_{d_{L}}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\rho\sigma}\,\delta_{ij}\right)\right]U_{\sigma\beta}\end{aligned}$ | 81 (45)
|  $\begin{aligned} &V^{\dagger}_{ik}\,\left[[{{C}}_{LL}^{V}]^{\alpha\beta kj}-\chi\,\left(k_{e\nu W}\,[\hat{{\mathbf{c}}}_{ud_{L}W}]^{kj}\,\delta_{\alpha\beta}+[k_{udW}]^{kj}\,[\hat{{\mathbf{c}}}_{e\nu_{L}W}]^{\alpha\beta}\right)\right]\\\ &=\left[[{{C}}_{edLL}^{V}]^{\alpha\rho ij}-\left(k_{e_{L}}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\alpha\rho}+k_{d_{L}}\,[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\rho}\,\delta_{ij}\right)\right]\,U^{\dagger}_{\rho\beta}\\\ &-U^{\dagger}_{\alpha\sigma}\,\left[[{{C}}_{\nu dLL}^{V}]^{\sigma\beta ij}-\left(k_{\nu_{L}}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\sigma\beta}+k_{d_{L}}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\sigma\beta}\,\delta_{ij}\right)\right]\end{aligned}$ | 162 (81)
$RRRR$ | No relations |
$LLRR$ |  $\begin{aligned} &[{{C}}_{edLR}^{V}]^{\alpha\beta ij}-\left(k_{e_{L}}\,[\hat{{\mathbf{c}}}_{d_{R}Z}]^{ij}\,\delta_{\alpha\beta}+k_{d_{R}}\,[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{ij}\right)\\\ &=U_{\alpha\rho}^{\dagger}\left[[{{C}}_{\nu dLR}^{V}]^{\rho\sigma ij}-\left(k_{\nu_{L}}\,[\hat{{\mathbf{c}}}_{d_{R}Z}]^{ij}\,\delta_{\rho\sigma}+k_{d_{R}}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\rho\sigma}\,\delta_{ij}\right)\right]\,U_{\sigma\beta}\end{aligned}$ | 81 (45)
|  $\begin{aligned} &[{{C}}_{euLR}^{V}]^{\alpha\beta ij}-\chi\,\left(k_{e_{L}}\,[\hat{{\mathbf{c}}}_{u_{R}Z}]^{ij}\,\delta_{\alpha\beta}+k_{u_{R}}\,[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{ij}\right)\\\ &=U^{\dagger}_{\alpha\rho}\,\left[[{{C}}_{\nu uLR}^{V}]^{\rho\sigma ij}-\chi\,\left(k_{\nu_{L}}\,[\hat{{\mathbf{c}}}_{u_{R}Z}]^{ij}\,\delta_{\rho\sigma}+k_{u_{R}}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\rho\sigma}\,\delta_{ij}\right)\right]\,U_{\sigma\beta}\end{aligned}$ | 81 (45)
|  $[\hat{{C}}_{LR}^{V}]^{\alpha\beta ij}=\chi\,k_{e\nu W}\,[\hat{{\mathbf{c}}}_{ud_{R}W}]^{ij}$ | 162 (81)
$RRLL$ |  $\begin{aligned} &[{{C}}_{edRL}^{V}]^{\alpha\beta ij}-\left(k_{e_{R}}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\alpha\beta}+k_{d_{L}}\,[\hat{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\beta}\,\delta_{ij}\right)\\\ &=V^{\dagger}_{ik}\,\left[[{{C}}_{euRL}^{V}]^{\alpha\beta kl}-\chi\,\left(k_{e_{R}}\,[\hat{{\mathbf{c}}}_{u_{L}Z}]^{kl}\,\delta_{\alpha\beta}+k_{u_{L}}\,[\hat{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\beta}\,\delta_{kl}\right)\right]\,V_{lj}\end{aligned}$ | 81 (45)
Table 8:  Relations among the LEFT semileptonic vector WCs and the BSM coupling of $Z,W^{\pm}$ and Higgs bosons to fermions, in the mass basis. Note that for the UV4f models, these BSM couplings vanish and this table becomes similar to Table 3, but in terms of only LEFT WCs. Class | Analytic relations for WCs of scalar operators | Count
---|---|---
Scalar ($d_{R}$) |  $\begin{aligned} &V_{ik}\,\left([{{C}}_{edRLLR}^{S}]^{\alpha\beta kj}-[\hat{{\mathbf{c}}}_{eh}]^{\beta\alpha\,*}{M}_{d}^{kj}\,k_{dh}-{M}_{e}^{\alpha\beta}\,k_{eh}[\hat{{\mathbf{c}}}_{dh}]^{kj}\right.\\\ &~{}-{M}_{e}^{\alpha\rho}\,[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\rho\beta}\,{M}_{d}^{kj}y_{d}+[\hat{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\rho}{M}_{e}^{\rho\beta}\,{M}_{d}^{kj}y_{d}\\\ &~{}\left.-y_{e}{M}_{e}^{\alpha\beta}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{km}{M}_{d}^{mj}+y_{e}{M}_{e}^{\alpha\beta}\,{M}_{d}^{km}\,[\hat{{\mathbf{c}}}_{d_{R}Z}]^{mj}\right)\\\ &=\left[[\hat{{C}}_{RLLR}^{S}]^{\alpha\rho ij}-\chi\left({M}_{e}^{\alpha\sigma}[\hat{{\mathbf{c}}}_{e\nu_{L}W}]^{\sigma\rho}\,M_{d}^{kj}y_{ud}\right.\right.\\\ &~{}\left.\left.+y_{e\nu}M_{e}^{\alpha\rho}\,[\hat{{\mathbf{c}}}_{ud_{L}W}]^{ik}{M}_{u}^{kj}-y_{e\nu}M_{e}^{\alpha\rho}\,{M}_{u}^{kj}[\hat{{\mathbf{c}}}_{ud_{R}W}]^{kj}\right)\right]\,U_{\rho\beta}\end{aligned}$ | 162 (81)
$\begin{aligned} &[{{C}}_{edRLRL}^{S}]^{\alpha\beta ij}=[\hat{{\mathbf{c}}}_{eh}]^{\beta\alpha\,*}\,{M}_{d}^{ij}\,k_{dh}+k_{eh}{M}_{e}^{\alpha\beta}\,[\hat{{\mathbf{c}}}_{dh}]^{ji\,*}\\\ &-{M}_{e}^{\alpha\rho}[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\rho\beta}\,{M}_{d}^{ij}y_{d}+[\hat{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\rho}{M}_{e}^{\rho\beta}\,{M}_{d}^{ij}y_{d}~{}\\\ &~{}-y_{e}{M}_{e}^{\alpha\beta}\,{M}_{d}^{ik}[\hat{{\mathbf{c}}}_{d_{L}Z}]^{kj}+y_{e}{M}_{e}^{\alpha\beta}\,[\hat{{\mathbf{c}}}_{d_{R}Z}]^{ik}\,{M}_{d}^{kj}\end{aligned}$ | 162 (81)
Scalar ($u_{R}$) |  $\begin{aligned} &\left[[{{C}}_{euRLRL}^{S}]^{\alpha\beta ik}-\chi\left([\hat{{\mathbf{c}}}_{eh}]^{\beta\alpha\,*}\,{M}_{u}^{ik}\,k_{uh}+k_{eh}{M}_{e}^{\alpha\beta}\,[\hat{{\mathbf{c}}}_{uh}]^{ki\,*}\right.\right.\\\ &-{M}_{e}^{\alpha\rho}[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\rho\beta}\,{M}_{u}^{ik}y_{u}+[\hat{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\rho}{M}_{e}^{\rho\beta}\,{M}_{u}^{ik}y_{u}~{}\\\ &~{}\left.\left.-y_{e}{M}_{e}^{\alpha\beta}\,{M}_{u}^{im}[\hat{{\mathbf{c}}}_{u_{L}Z}]^{mk}+y_{e}{M}_{e}^{\alpha\beta}\,[\hat{{\mathbf{c}}}_{u_{R}Z}]^{im}\,{M}_{uRL}^{mk}\right)\right]\,V_{kj}\\\ &=-\left[[{{C}}_{RLRL}^{S}]^{\alpha\rho ij}-\chi\left({M}_{e}^{\alpha\sigma}[\hat{{\mathbf{c}}}_{e\nu_{L}W}]^{\sigma\rho}\,M_{u}^{kj}y_{ud}\right.\right.\\\ &\left.\left.-y_{e\nu}M_{e}^{\alpha\rho}\,[\hat{{\mathbf{c}}}_{ud_{R}W}]^{ik}{M}_{d}^{kj}+y_{e\nu}M_{e}^{\alpha\rho}\,{M}_{u}^{kj}[\hat{{\mathbf{c}}}_{ud_{L}W}]^{kj}\right)\right]\,U_{\rho\beta}\end{aligned}$ | 162 (81)
$\begin{aligned} &[{{C}}_{euRLLR}^{S}]^{\alpha\beta ij}=\chi\left([\hat{{\mathbf{c}}}_{eh}]^{\beta\alpha\,*}{M}_{u}^{ij}\,k_{uh}+{M}_{e}^{\alpha\beta}\,k_{eh}[\hat{{\mathbf{c}}}_{uh}]^{ij}\right.\\\ &~{}+{M}_{e}^{\alpha\rho}\,[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\rho\beta}\,{M}_{u}^{ij}y_{u}-[\hat{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\rho}{M}_{e}^{\rho\beta}\,{M}_{u}^{ij}y_{u}\\\ &~{}\left.+y_{e}{M}_{e}^{\alpha\beta}\,[\hat{{\mathbf{c}}}_{u_{L}Z}]^{ik}{M}_{u}^{kj}-y_{e}{M}_{e}^{\alpha\beta}\,{M}_{u}^{ik}\,[\hat{{\mathbf{c}}}_{u_{R}Z}]^{kj}\right)\end{aligned}$ | 162 (81)
Tensor ($d_{R}$) | $[{{C}}_{ed,\,\textrm{all}}^{T}]^{\alpha\beta ij}=0$ | 324 (162)
$[{{C}}_{RLLR}^{T}]^{\alpha\beta ij}=0$ | 162 (81)
Tensor ($u_{R}$) | $[{{C}}_{eu,RLRL}^{T,\alpha\beta ik}\,V_{kj}=-[{{C}}_{RLRL}^{T}]^{\alpha\rho ij}\,U_{\rho\beta}$ | 162 (81)
$[{{C}}_{eu,LRRL}^{T}]^{\alpha\beta ij}=0$ | 162 (81)
Table 9:  Relations among the LEFT semileptonic scalar WCs and HEFT WCs for
the BSM coupling of $Z,W^{\pm}$ and Higgs bosons to fermions, in the mass
basis. Note that for the UV4f models, these BSM couplings vanish and this
table becomes similar to Table 3, but in terms of only LEFT WCs.

Now we present the matching relations among LEFT and HEFT operators in the
flavor basis.

For $LLLL$ vector operators:

$\displaystyle[\tilde{{C}}_{\nu uLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{\nu uLL}^{V}]^{\alpha\beta
ij}+\chi\,\left(k_{\nu_{L}}\,[{{\mathbf{c}}}_{u_{L}Z}]^{ij}\,\delta_{\alpha\beta}+k_{u_{L}}\,[{{\mathbf{c}}}_{\nu_{L}Z}]^{\alpha\beta}\,\delta_{ij}\right),$
(94) $\displaystyle[\tilde{{C}}_{euLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{euLL}^{V}]^{\alpha\beta
ij}+\chi\,\left(k_{e_{L}}\,[{{\mathbf{c}}}_{u_{L}Z}]^{ij}\,\delta_{\alpha\beta}+k_{u_{L}}\,[{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{ij}\right),$
(95) $\displaystyle[\tilde{{C}}_{\nu dLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{\nu dLL}^{V}]^{\alpha\beta
ij}+k_{\nu_{L}}\,[{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\alpha\beta}+k_{d_{L}}\,[{{\mathbf{c}}}_{\nu_{L}Z}]^{\alpha\beta}\,\delta_{ij},$
(96) $\displaystyle[\tilde{{C}}_{edLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{edLL}^{V}]^{\alpha\beta
ij}+k_{e_{L}}\,[{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\alpha\beta}+k_{d_{L}}\,[{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{ij},$
(97) $\displaystyle[\tilde{{C}}_{LL}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{LL}^{V}]^{\alpha\beta
ij}+\chi\,\left(k_{e\nu
W}\,[{{\mathbf{c}}}_{ud_{L}W}]^{ij}\,\delta_{\alpha\beta}+k_{udW}\,[{{\mathbf{c}}}_{e\nu_{L}W}]^{\alpha\beta}\,\delta_{ij}\right),$
(98)

Where $l\in\\{\nu,e\\}$, $q\in\\{u,d\\}$, $\omega=v^{2}/(2\,\Lambda^{2})$, and
the $k$ coefficients are

$\displaystyle k_{f_{L}}$
$\displaystyle=\frac{2\cos\theta_{w}}{g}(T^{3}_{f}-Q_{f}\sin^{2}\theta_{w})~{},\quad\textrm{and}~{}~{}k_{e\nu
W}=k_{udW}=\frac{\sqrt{2}}{g}~{},$ (99)

where $f_{L}\in\\{\nu_{L},e_{L},u_{L},d_{L}\\}$, as defined in Sec. 2.3.

For $LLRR$ vector operators:

$\displaystyle[\tilde{{C}}_{\nu uLR}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{\nu uLL}^{V}]^{\alpha\beta
ij}+\chi\,\left(k_{\nu_{L}}\,[{{\mathbf{c}}}_{u_{R}Z}]^{ij}\,\delta_{\alpha\beta}+k_{u_{R}}\,{{\mathbf{c}}}_{\nu_{L}Z}]^{\alpha\beta}\,\delta_{ij}\right),$
(100) $\displaystyle[\tilde{{C}}_{euLR}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{euLR}^{V}]^{\alpha\beta
ij}+\chi\,\left(k_{e_{L}}\,[{{\mathbf{c}}}_{u_{R}Z}]^{ij}\,\delta_{\alpha\beta}+k_{u_{R}}\,{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{ij}\right),$
(101) $\displaystyle[\tilde{{C}}_{\nu dLR}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{\nu dLR}^{V}]^{\alpha\beta
ij}+k_{\nu_{L}}\,[{{\mathbf{c}}}_{d_{R}Z}]^{ij}\,\delta_{\alpha\beta}+k_{d_{R}}\,[{{\mathbf{c}}}_{\nu_{L}Z}]^{\alpha\beta}\,\delta_{ij},$
(102) $\displaystyle[\tilde{{C}}_{edLR}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{edLR}^{V}]^{\alpha\beta
ij}+k_{e_{L}}\,[{{\mathbf{c}}}_{d_{R}Z}]^{ij}\,\delta_{\alpha\beta}+k_{d_{R}}\,[{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{ij},$
(103) $\displaystyle[\tilde{{C}}_{LR}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{LR}^{V}]^{\alpha\beta
ij}+\chi\,\left(k_{e\nu
W}\,[{{\mathbf{c}}}_{ud_{R}W}]^{ij}\,\delta_{\alpha\beta}\right).$ (104)

Here, $k_{e_{R}}$, $k_{u_{R}}$ and $k_{d_{R}}$ are

$\displaystyle k_{f_{R}}$
$\displaystyle=-\frac{2\cos\theta_{w}}{g}(Q_{f}\sin^{2}\theta_{w})~{}.$ (105)

For $RRRR$ vector operators:

$\displaystyle[\tilde{{C}}_{euRR}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{euRR}^{V}]^{\alpha\beta
ij}+\chi\,\left(k_{e_{R}}\,[{{\mathbf{c}}}_{u_{R}Z}]^{ij}\,\delta_{\alpha\beta}+k_{u_{R}}\,{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\beta}\,\delta_{ij}\right)~{},$
(106) $\displaystyle[\tilde{{C}}_{edRR}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{edRR}^{V}]^{\alpha\beta
ij}+k_{e_{R}}\,[{{\mathbf{c}}}_{d_{R}Z}]^{ij}\,\delta_{\alpha\beta}+k_{d_{R}}\,[{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\beta}\,\delta_{ij}~{}.$
(107)

In this category, the number of independent SMEFT and HEFT operators are the
same. As a result, there is no prediction for the LEFT WCs in this category.

For $RRLL$ vector operators:

$\displaystyle[\tilde{{C}}_{euRL}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{euRL}^{V}]^{\alpha\beta
ij}+\chi\,\left(k_{e_{R}}\,[{{\mathbf{c}}}_{u_{L}Z}]^{ij}\,\delta_{\alpha\beta}+k_{u_{L}}\,[{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\beta}\,\delta_{ij}\right),$
(108) $\displaystyle[\tilde{{C}}_{edRL}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{edRL}^{V}]^{\alpha\beta
ij}+k_{e_{R}}\,[{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\alpha\beta}+k_{d_{L}}\,[{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\beta}\,\delta_{ij}~{}.$
(109)

For scalar operators:

$\displaystyle[\tilde{{C}}_{edRLLR}^{S}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{edRLLR}^{S}]^{\alpha\beta
ij}+[{{\mathbf{c}}}_{eh}]^{\beta\alpha\,*}\tilde{M}_{dLR}^{ij}\,k_{dh}+\tilde{M}_{eRL}^{\alpha\beta}\,k_{eh}[{{\mathbf{c}}}_{dh}]^{ij}$
$\displaystyle~{}+\tilde{M}_{eRL}^{\alpha\rho}\,[{{\mathbf{c}}}_{e_{L}Z}]^{\rho\beta}\,\tilde{M}_{dLR}^{ij}y_{d}-[{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\rho}\tilde{M}_{eRL}^{\rho\beta}\,\tilde{M}_{dLR}^{ij}y_{d}$
$\displaystyle~{}+y_{e}\tilde{M}_{eRL}^{\alpha\beta}\,[{{\mathbf{c}}}_{d_{L}Z}]^{ik}\tilde{M}_{dRL}^{kj}-y_{e}\tilde{M}_{eRL}^{\alpha\beta}\,\tilde{M}_{dLR}^{ik}\,[{{\mathbf{c}}}_{d_{R}Z}]^{kj}~{},$
(110) $\displaystyle[\tilde{{C}}_{edRLRL}^{S}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{edRLRL}^{S}]^{\alpha\beta
ij}+[{{\mathbf{c}}}_{eh}]^{\beta\alpha\,*}\,\tilde{M}_{dRL}^{ij}\,k_{dh}+k_{eh}\tilde{M}_{eRL}^{\alpha\beta}\,[{{\mathbf{c}}}_{dh}]^{ji\,*}$
$\displaystyle-\tilde{M}_{eRL}^{\alpha\rho}[{{\mathbf{c}}}_{e_{L}Z}]^{\rho\beta}\,\tilde{M}_{dRL}^{ij}y_{d}+[{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\rho}\tilde{M}_{eLR}^{\rho\beta}\,\tilde{M}_{dRL}^{ij}y_{d}~{}$
$\displaystyle~{}-y_{e}\tilde{M}_{eRL}^{\alpha\beta}\,\tilde{M}_{dRL}^{ik}[{{\mathbf{c}}}_{d_{L}Z}]^{kj}+y_{e}\tilde{M}_{eRL}^{\alpha\beta}\,[{{\mathbf{c}}}_{d_{R}Z}]^{ik}\,\tilde{M}_{dRL}^{kj}~{},$
(111) $\displaystyle[\tilde{{C}}_{euRLLR}^{S}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{euRLLR}^{S}]^{\alpha\beta
ij}+\chi\left([{{\mathbf{c}}}_{eh}]^{\beta\alpha\,*}\tilde{M}_{uLR}^{ij}\,k_{uh}+\tilde{M}_{eRL}^{\alpha\beta}\,k_{eh}[{{\mathbf{c}}}_{uh}]^{ij}\right.$
$\displaystyle~{}+\tilde{M}_{eRL}^{\alpha\rho}\,[{{\mathbf{c}}}_{e_{L}Z}]^{\rho\beta}\,\tilde{M}_{uLR}^{ij}y_{u}-[{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\rho}\tilde{M}_{eRL}^{\rho\beta}\,\tilde{M}_{uLR}^{ij}y_{u}$
$\displaystyle~{}\left.+y_{e}\tilde{M}_{eRL}^{\alpha\beta}\,[{{\mathbf{c}}}_{u_{L}Z}]^{ik}\tilde{M}_{uRL}^{kj}-y_{e}\tilde{M}_{eRL}^{\alpha\beta}\,\tilde{M}_{uLR}^{ik}\,[{{\mathbf{c}}}_{u_{R}Z}]^{kj}\right)~{},$
(112) $\displaystyle[\tilde{{C}}_{euRLRL}^{S}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{euRLRL}^{S}]^{\alpha\beta
ij}+\chi\left([{{\mathbf{c}}}_{eh}]^{\beta\alpha\,*}\,\tilde{M}_{uRL}^{ij}\,k_{uh}+k_{eh}\tilde{M}_{eRL}^{\alpha\beta}\,[{{\mathbf{c}}}_{uh}]^{ji\,*}\right.$
$\displaystyle-\tilde{M}_{eRL}^{\alpha\rho}[{{\mathbf{c}}}_{e_{L}Z}]^{\rho\beta}\,\tilde{M}_{uRL}^{ij}y_{u}+[{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\rho}\tilde{M}_{eLR}^{\rho\beta}\,\tilde{M}_{uRL}^{ij}y_{u}~{}$
$\displaystyle~{}\left.-y_{e}\tilde{M}_{eRL}^{\alpha\beta}\,\tilde{M}_{uRL}^{ik}[{{\mathbf{c}}}_{u_{L}Z}]^{kj}+y_{e}\tilde{M}_{eRL}^{\alpha\beta}\,[{{\mathbf{c}}}_{u_{R}Z}]^{ik}\,\tilde{M}_{uRL}^{kj}\right)~{},$
(113) $\displaystyle[\tilde{{C}}_{RLLR}^{S}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{RLLR}^{S}]^{\alpha\beta
ij}+\chi\left(\tilde{M}_{eRL}^{\alpha\rho}[{{\mathbf{c}}}_{e\nu_{L}W}]^{\rho\beta}\,\tilde{M}_{dLR}^{ij}y_{ud}+y_{e\nu}\tilde{M}_{eRL}^{\alpha\beta}\,[{{\mathbf{c}}}_{ud_{L}W}]^{ik}\tilde{M}_{dLR}^{kj}\right.$
$\displaystyle~{}\left.~{}-y_{e\nu}\tilde{M}_{eRL}^{\alpha\beta}\,\tilde{M}_{uLR}^{ik}[{{\mathbf{c}}}_{ud_{R}W}]^{kj}\right)~{},$
(114) $\displaystyle[\tilde{{C}}_{RLRL}^{S}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{RLRL}^{S}]^{\alpha\beta
ij}+\chi\left(\tilde{M}_{eRL}^{\alpha\rho}[{{\mathbf{c}}}_{e\nu_{L}W}]^{\rho\beta}\,\tilde{M}_{uRL}^{ij}y_{ud}-y_{e\nu}\tilde{M}_{eRL}^{\alpha\beta}\,[{{\mathbf{c}}}_{ud_{R}W}]^{ik}\tilde{M}_{dRL}^{kj}\right.$
$\displaystyle~{}\left.~{}+y_{e\nu}\tilde{M}_{eRL}^{\alpha\beta}\,\tilde{M}_{uRL}^{ik}[{{\mathbf{c}}}_{ud_{L}W}]^{kj}\right)~{},$
(115)

where

$\displaystyle\tilde{M}_{uLR}$
$\displaystyle=S^{u}_{L}\,M_{u}\,S^{u\dagger}_{R},\quad\tilde{M}_{dLR}=S^{d}_{L}\,M_{d}\,S^{d\dagger}_{R},\quad\tilde{M}_{eLR}=K^{e}_{L}\,M_{e}\,K^{e\dagger}_{R}~{},$
(116)

with $M_{e}={\rm diag}(m_{e},~{}m_{\mu},~{}m_{\tau})$, $M_{u}={\rm
diag}(m_{u},~{}m_{c},~{}m_{t})$, and $M_{d}={\rm
diag}(m_{d},~{}m_{s},~{}m_{b})$. For $\tilde{M}_{fRL}$, $L$ and $R$ are
interchanged compared to the $\tilde{M}_{fLR}$ expressions in eq. (116). The
$k$-coefficients are

$\displaystyle k_{eh}$ $\displaystyle=k_{uh}=k_{dh}=\frac{v}{2m_{h}^{2}}~{}.$
(117)

Note that the BSM couplings of $W^{\pm}$ and $Z$ bosons can also contribute to
scalar LEFT operators because of the second term in the unitary gauge
propagator for massive vectors. In the matching relations above for the $Z$
and $W^{\pm}$ couplings to fermions, the corresponding coefficients are

$\displaystyle y_{f}$
$\displaystyle=\frac{1}{m_{Z}^{2}}(k_{f_{L}Z}-k_{f_{R}Z})~{},\quad
y_{ud}=y_{e\nu}=\frac{k_{udW}}{m_{W}^{2}}~{}.$ (118)

Note that in the UV4f scenario, the vanishing of HEFT WCs
${{\mathbf{c}}}_{edLRLR}^{S}$ and ${{\mathbf{c}}}_{euLRRL}^{S}$ results in the
vanishing of the LEFT WCs ${{C}}_{edLRLR}^{S}$ and ${{C}}_{euLRRL}^{S}$. In
conventional LEFT notation Aebischer:2015fzz , these identities are presented
as $C_{S}=-C_{P}$ and $C_{S}^{\prime}=C_{P}^{\prime}$ in both of these
categories.

For tensor operators:

$\displaystyle[\tilde{{C}}_{edLRRL}^{T}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{edLRRL}^{T}]^{\alpha\beta
ij}~{},\quad[\tilde{{C}}_{edLRLR}^{T}]^{\alpha\beta
ij}=\omega\,[{{\mathbf{c}}}_{edLRRL}^{T}]^{\alpha\beta ij}~{},$ (119)
$\displaystyle[\tilde{{C}}_{euLRRL}^{T}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{euLRRL}^{T}]^{\alpha\beta
ij}~{},\quad[\tilde{{C}}_{euLRLR}^{T}]^{\alpha\beta
ij}=\omega\,[{{\mathbf{c}}}_{euLRRL}^{T}]^{\alpha\beta ij}~{},$ (120)
$\displaystyle[\tilde{{C}}_{RLLR}^{T}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{RLLR}^{T}]^{\alpha\beta
ij}\quad\textrm{, and}\quad[\tilde{{C}}_{RLRL}^{T}]^{\alpha\beta
ij}=\omega\,[{{\mathbf{c}}}_{RLRL}^{T}]^{\alpha\beta ij}~{}.$ (121)

The relations among the tensor LEFT WCs remain the same as for the HEFT WCs.

Note that, while writing the relations in Tables 8 and 9, we have replaced the
four-fermion HEFT WCs with the corresponding LEFT ones using the matching
relations. However, when the high energy bounds for the four-fermion WCs are
stronger than the low energy bounds, it may be more efficient to use some of
the original HEFT WCs instead of replacing them with the LEFT ones.

##  Appendix D Tables for direct and indirect bounds on the WCs

In this section, we have listed the $90\%$ C.L. bounds on the WCs (on both
their real and the imaginary parts if they are complex ) obtained in Sec. 3.
The actual bounds especially between real and imaginary parts of the WCs are
correlated and the figures shown in sec. 3 are a better representation of the
bounds. However, for the sake of simplicity, we present the $90\%$ C.L. bounds
for individual quantities.

In Tables 10 to 13, we present the direct bounds derived from low-energy
observables (e.g. meson decays, neutrino oscillation), high-energy observables
(e.g. high-$p_{T}$ dimuon searches, production and decays of top quark) and
indirect bounds derived using eqs. (43) and (44). In Table 14, we present the
direct bounds from high-$p_{T}$ monomuon searches and the indirect bounds
derived from eq. (45). Note that the high-energy bounds directly constrain
only the HEFT WCs, $[\hat{{\mathbf{c}}}]^{\alpha\beta ij}$, but here we work
in the context of the UV4f scenario, which allows us to provide high-energy
bounds on LEFT WCs.

WC | Low energy | High energy | Indirect
---|---|---|---
$[{{C}}_{\nu dLL}^{V}]^{2211}$ | [-0.042, 0.049] | - | [-0.00021, 0.00046]
$\textrm{Re}\left([{{C}}_{\nu dLL}^{V}]^{2212}\right)$ | [-0.00001, 0.00001] | - | -
$\textrm{Im}\left([{{C}}_{\nu dLL}^{V}]^{2212}\right)$ | [-0.00001, 0.00001] | - | -
$\textrm{Re}\left([{{C}}_{\nu dLL}^{V}]^{2213}\right)$ | [-0.0010, 0.0012] | - | -
$\textrm{Im}\left([{{C}}_{\nu dLL}^{V}]^{2213}\right)$ | [-0.0012, 0.0011] | - | -
$[{{C}}_{\nu dLL}^{V}]^{2222}$ | - | - | [-0.0029, 0.0032]
$\textrm{Re}\left([{{C}}_{\nu dLL}^{V}]^{2223}\right)$ | [-0.0014, 0.00078] | - | -
$\textrm{Im}\left([{{C}}_{\nu dLL}^{V}]^{2223}\right)$ | [-0.0011, 0.0011] | - | -
$[{{C}}_{\nu dLL}^{V}]^{2233}$ | - | - | [-4.34, 4.44]
Table 10: $90\%$ C.L. bounds for the WCs of the type ${{C}}_{\nu dLL}^{V}$. The low-energy bounds (second column) correspond to bounds from rare decays of $K$ and $B$ mesons. For $[{{C}}_{\nu dLL}^{V}]^{2211}$, the low-energy bound is from atmospheric and accelerator neutrino experiments. There is no high-energy bound on these WCs. The fourth column shows the indirect bounds derived using eq. (43). WC | Low energy | High energy | Indirect
---|---|---|---
$[{{C}}_{euLL}^{V}]^{2211}$ | - | [-0.000077, 0.00033] | -
$\textrm{Re}\left([{{C}}_{euLL}^{V}]^{2212}\right)$ | [-0.00056, 0.00056] | [-0.00091, 0.00091] | -
$\textrm{Im}\left([{{C}}_{euLL}^{V}]^{2212}\right)$ | [-0.00056, 0.00056] | [-0.00091, 0.00091] | -
$\textrm{Re}\left([{{C}}_{euLL}^{V}]^{2213}\right)$ | - | [-0.0025, 0.0025] | [-0.0078, 0.0077]
$\textrm{Im}\left([{{C}}_{euLL}^{V}]^{2213}\right)$ | - | [-0.0025, 0.0025] | [-0.013, 0.013]
$[{{C}}_{euLL}^{V}]^{2222}$ | - | [-0.0039, 0.00093] | [-0.0067, 0.0071]
$\textrm{Re}\left([{{C}}_{euLL}^{V}]^{2223}\right)$ | - | [-0.0208, 0.0208] | [-0.18, 0.19]
$\textrm{Im}\left([{{C}}_{\nu dLL}^{V}]^{2223}\right)$ | - | [-0.021, 0.021] | [-0.0040, 0.0041]
$[{{C}}_{euLL}^{V}]^{2233}$ | - | [-0.20, 0.20] | [-4.34, 4.44]
Table 11: $90\%$ C.L. bounds for the WCs of the type ${{C}}_{euLL}^{V}$. The low-energy bounds (second column) correspond to bounds from rare $D$ meson decays. The high-energy bounds (third column) correspond to bounds derived from high-$p_{T}$ dimuon searches and from top production and decays, particularly for $[{{C}}_{euLL}^{V}]^{2213}$, $[{{C}}_{euLL}^{V}]^{2223}$ and $[{{C}}_{euLL}^{V}]^{2233}$. The fourth column shows the indirect bounds derived using eq. (43). WC | Low energy | High energy | Indirect
---|---|---|---
$[{{C}}_{edLL}^{V}]^{2211}$ | - | [-0.0010, 0.00014] | -
$\textrm{Re}\left([{{C}}_{edLL}^{V}]^{2212}\right)$ | [-0.00001, 0.00001] | [-0.0011, 0.0011] | -
$\textrm{Im}\left([{{C}}_{edLL}^{V}]^{2212}\right)$ | [-0.00001, 0.00001] | [-0.0011, 0.0011] | -
$\textrm{Re}\left([{{C}}_{edLL}^{V}]^{2213}\right)$ | [-0.00003, 0.00001] | [-0.0017, 0.0017] | -
$\textrm{Im}\left([{{C}}_{edLL}^{V}]^{2213}\right)$ | [-0.00002, 0.00003] | [-0.0017, 0.0017] | -
$[{{C}}_{edLL}^{V}]^{2222}$ | - | [-0.0043, 0.0014] | -
$\textrm{Re}\left([{{C}}_{edLL}^{V}]^{2223}\right)$ | [-0.00013, 0.00010] | [-0.0041, 0.0041] | -
$\textrm{Im}\left([{{C}}_{edLL}^{V}]^{2223}\right)$ | [-0.00017, 0.00017] | [-0.0041, 0.0041] | -
$[{{C}}_{edLL}^{V}]^{2233}$ | - | [-0.0096, 0.0054] | -
Table 12: $90\%$ C.L. bounds for the WCs of the type ${{C}}_{edLL}^{V}$. The low-energy bounds (second column) correspond to bounds from rare decays of $K$ and $B$ mesons. High-energy bounds (third column) correspond to bounds derived from high-$p_{T}$ dimuon searches. The indirect bounds are absent as we have used these WCs as inputs to eq. (44). WC | Low energy | High energy | Indirect
---|---|---|---
$[{{C}}_{\nu uLL}^{V}]^{2211}$ | [-0.075, 0.052] | - | [-0.0012, 0.00021]
$\textrm{Re}\left([{{C}}_{\nu uLL}^{V}]^{2212}\right)$ | - | - | [-0.00098, 0.00053]
$\textrm{Im}\left([{{C}}_{\nu uLL}^{V}]^{2212}\right)$ | - | - | [-0.000012, 0.000013]
$\textrm{Re}\left([{{C}}_{\nu uLL}^{V}]^{2213}\right)$ | - | - | [-0.000057, 0.000074]
$\textrm{Im}\left([{{C}}_{\nu uLL}^{V}]^{2213}\right)$ | - | - | [-0.000070, 0.000092]
$[{{C}}_{\nu uLL}^{V}]^{2222}$ | - | - | [-0.0042, 0.0013]
$\textrm{Re}\left([{{C}}_{\nu uLL}^{V}]^{2223}\right)$ | - | - | [-0.00044, 0.00049]
$\textrm{Im}\left([{{C}}_{\nu uLL}^{V}]^{2223}\right)$ | - | - | [-0.00017, 0.00017]
$[{{C}}_{\nu uLL}^{V}]^{2233}$ | - | - | [-0.0096, 0.0054]
Table 13: $90\%$ C.L. bounds for the WCs of the type ${{C}}_{\nu uLL}^{V}$.The low-energy bound (second column) for $[{{C}}_{\nu uLL}^{V}]^{2211}$ is obtained from atmospheric and accelerator neutrino experiments. There is no high-energy bound on these WCs. The fourth column shows the indirect bounds derived using eq. (44). WC |  | High energy | Indirect
---|---|---|---
$\textrm{Re}\left([{{C}}_{LL}^{V}]^{2211}\right)$ |  | [-0.0006, 0.0002] | [-0.0012, 0.00028]
$\textrm{Im}\left([{{C}}_{LL}^{V}]^{2211}\right)$ |  | [-0.0014, 0.0014] | [-0.0000070, 0.0000078]
$\textrm{Re}\left([{{C}}_{LL}^{V}]^{2212}\right)$ |  | [-0.0016, 0.0009] | [-0.0015, 0.00084]
$\textrm{Im}\left([{{C}}_{LL}^{V}]^{2212}\right)$ |  | [-0.0013, 0.0013] | [-0.000024, 0.000019]
$\textrm{Re}\left([{{C}}_{LL}^{V}]^{2213}\right)$ |  | [-0.0017, 0.0013] | [-0.0015, 0.0015]
$\textrm{Im}\left([{{C}}_{LL}^{V}]^{2213}\right)$ |  | [-0.0015, 0.0016] | [-0.0018, 0.0018]
$\textrm{Re}\left([{{C}}_{LL}^{V}]^{2221}\right)$ |  | [-0.0014, 0.0022] | [-0.00010, 0.00033]
$\textrm{Im}\left([{{C}}_{LL}^{V}]^{2221}\right)$ |  | [-0.0018, 0.0018] | [-0.000056, 0.000059]
$\textrm{Re}\left([{{C}}_{LL}^{V}]^{2222}\right)$ |  | [-0.006, 0.002] | [-0.0066, 0.0037]
$\textrm{Im}\left([{{C}}_{LL}^{V}]^{2222}\right)$ |  | [-0.0043, 0.0043] | [-0.000052, 0.000054]
$\textrm{Re}\left([{{C}}_{LL}^{V}]^{2223}\right)$ |  | [-0.005, 0.005] | [-0.0095, 0.0098]
$\textrm{Im}\left([{{C}}_{LL}^{V}]^{2223}\right)$ |  | [-0.005, 0.005] | [-0.0014, 0.0014]
$\textrm{Re}\left([{{C}}_{LL}^{V}]^{2231}\right)$ |  | - | [-0.0012, 0.0010]
$\textrm{Im}\left([{{C}}_{LL}^{V}]^{2231}\right)$ |  | - | [-0.0011, 0.0011]
$\textrm{Re}\left([{{C}}_{LL}^{V}]^{2232}\right)$ |  | - | [-0.00087, 0.0017]
$\textrm{Im}\left([{{C}}_{LL}^{V}]^{2232}\right)$ |  | - | [-0.0012, 0.0012]
$\textrm{Re}\left([{{C}}_{LL}^{V}]^{2233}\right)$ |  | - | [-0.21, 0.20]
$\textrm{Im}\left([{{C}}_{LL}^{V}]^{2233}\right)$ |  | - | [-0.000059, 0.000059]
Table 14: $90\%$ C.L. bounds for the charged-current WCs of the type
${{C}}_{LL}^{V}$. The high-energy bounds (second column) are derived from
high-$p_{T}$ mono-muon searches. The third column shows the indirect bounds
derived using eq. (45).

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