Ramsey's theorem

There is an analogue of Ramsey's theorem from combinatorics for polyadic spaces. For this, we describe the relationship between Boolean spaces and polyadic spaces. Let ${\displaystyle CO(X)}$ denote the clopen algebra of all clopen subsets of ${\displaystyle X}$. We define a Boolean space as a compact Hausdorff space whose basis is ${\displaystyle CO(X)}$. The element ${\displaystyle G\in CO(X)'}$ such that ${\displaystyle \langle \langle G\rangle \rangle =CO(X)}$ is called the generating set for ${\displaystyle CO(X)}$. We say ${\displaystyle G}$ is a ${\displaystyle (\tau ,\kappa )}$-disjoint collection if ${\displaystyle G}$ is the union of at most ${\displaystyle \tau }$ subcollections ${\displaystyle G_{\alpha }}$, where for each ${\displaystyle \alpha }$, ${\displaystyle G_{\alpha }}$ is a disjoint collection of cardinality at most ${\displaystyle \kappa }$ It was proven by Petr Simon that ${\displaystyle X}$ is a Boolean space with the generating set ${\displaystyle G}$ of ${\displaystyle CO(X)}$ being ${\displaystyle (\tau ,\kappa )}$-disjoint if and only if ${\displaystyle X}$ is homeomorphic to a closed subspace of ${\displaystyle \alpha \kappa ^{\tau }}$.^[8] The Ramsey-like property for polyadic spaces as stated by Murray Bell for Boolean spaces is then as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint.^[13]

Compactness

We define the compactness number of a space ${\displaystyle X}$, denoted by ${\displaystyle \operatorname {cmpn} \,X}$, to be the least number ${\displaystyle n}$ such that ${\displaystyle X}$ has an n-ary closed subbase. We can construct polyadic spaces with arbitrary compactness number. We will demonstrate this using two theorems proven by Murray Bell in 1985. Let ${\displaystyle {\mathcal {S}}}$ be a collection of sets and let ${\displaystyle S}$ be a set. We denote the set ${\displaystyle \{\bigcap {\mathcal {F}}:{\mathcal {F}}{\text{ is a finite subset of }}{\mathcal {S}}\}}$ by ${\displaystyle {\mathcal {S}}^{\widehat {\mathcal {F}}}}$; all subsets of ${\displaystyle S}$ of size ${\displaystyle n}$ by ${\displaystyle [S]^{n}}$; and all subsets of size at most ${\displaystyle n}$ by ${\displaystyle [S]^{<=n}}$. If ${\displaystyle 2\leq n<\omega }$ and ${\displaystyle \bigcap {\mathcal {F}}\neq \emptyset }$ for all ${\displaystyle {\mathcal {F}}\in [{\mathcal {S}}]^{n}}$, then we say that ${\displaystyle {\mathcal {S}}}$ is n-linked. If every n-linked subset of ${\displaystyle {\mathcal {S}}}$ has a non-empty intersection, then we say that ${\displaystyle {\mathcal {S}}}$ is n-ary. Note that if ${\displaystyle {\mathcal {S}}}$ is n-ary, then so is ${\displaystyle {\mathcal {S}}^{\widehat {\mathcal {F}}}}$, and therefore every space ${\displaystyle X}$ with ${\displaystyle \operatorname {cmpn} \,X\leq n}$ has a closed, n-ary subbase ${\displaystyle {\mathcal {S}}}$ with ${\displaystyle {\mathcal {S}}={\mathcal {S}}^{\widehat {\mathcal {F}}}}$. Note that a collection ${\displaystyle {\mathcal {S}}={\mathcal {S}}^{\widehat {\mathcal {F}}}}$ of closed subsets of a compact space ${\displaystyle X}$ is a closed subbase if and only if for every closed ${\displaystyle K}$ in an open set ${\displaystyle U}$, there exists a finite ${\displaystyle {\mathcal {F}}}$ such that ${\displaystyle {\mathcal {F}}\subset {\mathcal {S}}}$ and ${\displaystyle K\subset \bigcup {\mathcal {F}}\subset U}$.^[14]

Let ${\displaystyle S}$ be an infinite set and let ${\displaystyle n}$ by a number such that ${\displaystyle 1\leq n<\omega }$. We define the product topology on ${\displaystyle [S]^{\leq n}}$ as follows: for ${\displaystyle s\in S}$, let ${\displaystyle s^{-}=\{F\in [S]^{\leq n}:s\in F\}}$, and let ${\displaystyle s^{+}=\{F\in [S]^{\leq n}:s\notin F\}}$. Let ${\displaystyle {\mathcal {S}}}$ be the collection ${\displaystyle {\mathcal {S}}=\bigcup _{s\in S}\{s^{+},s^{-}\}}$. We take ${\displaystyle {\mathcal {S}}}$ as a clopen subbase for our topology on ${\displaystyle [S]^{\leq n}}$. This topology is compact and Hausdorff. For ${\displaystyle k}$ and ${\displaystyle n}$ such that ${\displaystyle 0\leq k\leq n}$, we have that ${\displaystyle [S]^{k}}$ is a discrete subspace of ${\displaystyle [S]^{\leq n}}$, and hence that ${\displaystyle [S]^{\leq n}}$ is a union of ${\displaystyle n+1}$ discrete subspaces.^[14]

Theorem (Upper bound on ${\displaystyle \operatorname {cmpn} \,[S]^{\leq n}}$): For each total order ${\displaystyle <}$ on ${\displaystyle S}$, there is an ${\displaystyle n+1}$-ary closed subbase ${\displaystyle {\mathcal {R}}}$ of ${\displaystyle [S]^{\leq 2n}}$.

Proof: For ${\displaystyle s\in S}$, define ${\displaystyle L_{s}=\{F\in s^{+}:|\{t\in F:t<s\}|\leq n-1\}}$ and ${\displaystyle R_{s}=\{F\in s^{+}:|\{t\in F:t>s\}|\leq n-1\}}$. Set ${\displaystyle {\mathcal {R}}=\bigcup _{s\in S}\{L_{s},R_{s},s^{+}\}}$. For ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$ such that ${\displaystyle A\cup B\cup C\neq \emptyset }$, let ${\displaystyle {\mathcal {F}}=\{L_{s}:s\in A\}\cup \{R_{s}:s\in B\}\cup \{s^{-}:s\in C\}}$ such that ${\displaystyle {\mathcal {F}}}$ is an ${\displaystyle n+1}$-linked subset of ${\displaystyle {\mathcal {R}}}$. Show that ${\displaystyle A\cup B\in \bigcap {\mathcal {F}}}$. ${\displaystyle \blacksquare }$

For a topological space ${\displaystyle X}$ and a subspace ${\displaystyle A\in X}$, we say that a continuous function ${\displaystyle r:X\rightarrow A}$ is a retraction if ${\displaystyle r|_{A}}$ is the identity map on ${\displaystyle A}$. We say that ${\displaystyle A}$ is a retract of ${\displaystyle X}$. If there exists an open set ${\displaystyle U}$ such that ${\displaystyle A\subset U\subset X}$, and ${\displaystyle A}$ is a retract of ${\displaystyle U}$, then we say that ${\displaystyle A}$ is a neighbourhood retract of ${\displaystyle X}$.

Theorem (Lower bound on ${\displaystyle \operatorname {cmpn} \,[S]^{\leq n}}$) Let ${\displaystyle n}$ be such that ${\displaystyle 2\leq n<\omega }$. Then ${\displaystyle [\omega _{1}]^{\leq 2n-1}}$ cannot be embedded as a neighbourhood retract in any space ${\displaystyle K}$ with ${\displaystyle \operatorname {cmpn} \,K\leq n}$.

From the two theorems above, it can be deduced that for ${\displaystyle n}$ such that ${\displaystyle 1\leq n<\omega }$, we have that ${\displaystyle \operatorname {cmpn} \,[\omega _{1}]^{\leq 2n-1}=n+1=\operatorname {cmpn} \,[\omega _{1}]^{\leq 2n}}$.

Let ${\displaystyle A}$ be the Alexandroff one-point compactification of the discrete space ${\displaystyle S}$, so that ${\displaystyle A=S\cup \{\infty \}}$. We define the continuous surjection ${\displaystyle g:A^{n}\rightarrow [S]^{\leq n}}$ by ${\displaystyle g((x_{1},...,x_{n}))=\{x_{1},\ldots ,x_{n}\}\cap S}$. It follows that ${\displaystyle [S]^{\leq n}}$ is a polyadic space. Hence ${\displaystyle [\omega _{1}]^{\leq 2n-1}}$ is a polyadic space with compactness number ${\displaystyle \operatorname {cmpn} \,[\omega _{1}]^{\leq 2n-1}=n+1}$.^[14]

Centred space

Let ${\displaystyle {\mathcal {S}}}$ be a collection of sets. We say that ${\displaystyle {\mathcal {S}}}$ is centred if ${\displaystyle \bigcap {\mathcal {F}}\neq \emptyset }$ for all finite subsets ${\displaystyle {\mathcal {F}}\subseteq {\mathcal {S}}}$.^[17] Define the Boolean space ${\displaystyle Cen({\mathcal {S}})=\{\chi _{T}:T{\text{ is a centred subcollection of }}{\mathcal {S}}\}}$, with the subspace topology from ${\displaystyle 2^{\mathcal {S}}}$. We say that a space ${\displaystyle X}$ is a centred space if there exists a collection ${\displaystyle {\mathcal {S}}}$ such that ${\displaystyle X}$ is a continuous image of ${\displaystyle Cen({\mathcal {S}})}$.^[18]

Centred spaces were introduced by Murray Bell in 2004.

AD-compact space

Let ${\displaystyle X}$ be a non-empty set, and consider a family of its subsets ${\displaystyle {\mathcal {A}}\subseteq {\mathcal {P}}(X)}$. We say that ${\displaystyle {\mathcal {A}}}$ is an adequate family if:

• ${\displaystyle A\in {\mathcal {A}}\land B\subseteq {\mathcal {A}}\Rightarrow B\in {\mathcal {A}}}$
• given ${\displaystyle A\subseteq X}$, if every finite subset of ${\displaystyle A}$ is in ${\displaystyle {\mathcal {A}}}$, then ${\displaystyle A\in {\mathcal {A}}}$.

We may treat ${\displaystyle {\mathcal {A}}}$ as a topological space by considering it a subset of the Cantor cube ${\displaystyle D^{X}}$, and in this case, we denote it ${\displaystyle K({\mathcal {A}})}$.

Let ${\displaystyle K}$ be a compact space. If there exist a set ${\displaystyle X}$ and an adequate family ${\displaystyle {\mathcal {A}}\subseteq {\mathcal {P}}(X)}$, such that ${\displaystyle K}$ is the continuous image of ${\displaystyle K({\mathcal {A}})}$, then we say that ${\displaystyle K}$ is an AD-compact space.

AD-compact spaces were introduced by Grzegorz Plebanek. He proved that they are closed under arbitrary products and Alexandroff compactifications of disjoint unions. It follows that every polyadic space is hence an AD-compact space. The converse is not true, as there are AD-compact spaces that are not polyadic.^[15]

ξ-adic space

Let ${\displaystyle \kappa }$ and ${\displaystyle \tau }$ be cardinals, and let ${\displaystyle X}$ be a Hausdorff space. If there exists a continuous surjection from ${\displaystyle (\kappa +1)^{\tau }}$ to ${\displaystyle X}$, then ${\displaystyle X}$ is said to be a ξ-adic space.^[16]

ξ-adic spaces were proposed by S. Mrówka, and the following results about them were given by János Gerlits (they also apply to polyadic spaces, as they are a special case of ξ-adic spaces).^[19]

Let ${\displaystyle {\mathfrak {n}}}$ be an infinite cardinal, and let ${\displaystyle X}$ be a topological space. We say that ${\displaystyle X}$ has the property ${\displaystyle \mathbf {B} ({\mathfrak {n}})}$ if for any family ${\displaystyle \{G_{\alpha }:\alpha \in A\}}$ of non-empty open subsets of ${\displaystyle X}$, where ${\displaystyle |A|={\mathfrak {n}}}$, we can find a set ${\displaystyle B\subset A}$ and a point ${\displaystyle p\in X}$ such that ${\displaystyle |B|={\mathfrak {n}}}$ and for each neighbourhood ${\displaystyle N}$ of ${\displaystyle p}$, we have that ${\displaystyle |\{\beta \in B:N\cap G_{\beta }=\emptyset \}|<{\mathfrak {n}}}$.

If ${\displaystyle X}$ is a ξ-adic space, then ${\displaystyle X}$ has the property ${\displaystyle \mathbf {B} ({\mathfrak {n}})}$ for each infinite cardinal ${\displaystyle {\mathfrak {n}}}$. It follows from this result that no infinite ξ-adic Hausdorff space can be an extremally disconnected space.^[19]

Hyadic space

Hyadic spaces were introduced by Eric van Douwen.^[20] They are defined as follows.

Let ${\displaystyle X}$ be a Hausdorff space. We denote by ${\displaystyle H(X)}$ the hyperspace of ${\displaystyle X}$. We define the subspace ${\displaystyle J_{2}(X)}$ of ${\displaystyle H(X)}$ by ${\displaystyle \{F\in H(X):|F|\leq 2\}}$. A base of ${\displaystyle H(X)}$ is the family of all sets of the form ${\displaystyle \langle U_{0},\dots ,U_{n}\rangle =\{F\in H(X):F\subseteq U_{0}\cup \dots \cup U_{n},F\cap U_{i}\neq \emptyset {\text{ for }}0\leq i\leq n\}}$, where ${\displaystyle n}$ is any integer, and ${\displaystyle U_{i}}$ are open in ${\displaystyle X}$. If ${\displaystyle X}$ is compact, then we say a Hausdorff space ${\displaystyle Y}$ is hyadic if there exists a continuous surjection from ${\displaystyle H(X)}$ to ${\displaystyle Y}$.^[21]

Polyadic spaces are hyadic.^[22]