Polyadic space

Polyadic spaces were first studied by S. Mrówka in 1970 as a generalisation of dyadic spaces.{{cite book|title=Encyclopedia of General Topology|url=https://archive.org/details/encyclopediagene00hart|url-access=limited|chapter=Dyadic compacta|first1=Klaas Pieter|last1=Hart|author-link1=Klaas Pieter Hart|first2=Jun-iti|last2=Nagata|author-link2=Jun-iti Nagata|first3=Jerry E.|last3=Vaughan|page=[https://archive.org/details/encyclopediagene00hart/page/n185 193]|publisher=Elsevier Science|isbn=978-0444503558|date=2003}} The theory was developed further by R. H. Marty, János Gerlits and Murray G. Bell, the latter of whom introduced the concept of the more general centred spaces.

A subset K of a topological space X is said to be compact if every open cover of K contains a finite subcover. It is said to be locally compact at a point x ∈ X if x lies in the interior of some compact subset of X. X is a locally compact space if it is locally compact at every point in the space.{{cite book|last=Møller|first=Jesper M.|page=58|chapter=Topological spaces and continuous maps|title=General Topology|isbn=9781502795878|date=2014|publisher=CreateSpace Independent Publishing Platform }}

A proper subset A ⊂ X is said to be dense if the closure Ā = X. A space whose set has a countable, dense subset is called a separable space.

For a non-compact, locally compact Hausdorff topological space $(X,\; \backslash tau\_X)$, we define the Alexandroff one-point compactification as the topological space with the set $\backslash left\; \backslash \{\; \backslash omega\; \backslash right\; \backslash \}\; \backslash cup\; X$, denoted $\backslash omega\; X$, where $\backslash omega\; \backslash notin\; X$, with the topology $\backslash tau\_\{\backslash omega\; X\}$ defined as follows:{{cite book|title=A Cp-Theory Problem Book: Topological and Function Spaces|url=https://archive.org/details/cptheoryproblemb00vvtk|url-access=limited|chapter=Basic Notions of Topology and Function Spaces|first=Vladimir V.|last=Tkachuk|page=[https://archive.org/details/cptheoryproblemb00vvtk/page/n51 35]|isbn=9781441974426|date=2011|publisher=Springer Science+Business Media}}

• $\backslash tau\_X\; \backslash subseteq\; \backslash tau\_\{\backslash omega\; X\}$
• $X\; \backslash setminus\; C\; \backslash cup\; \backslash left\; \backslash \{\; \backslash omega\; \backslash right\; \backslash \}\; \backslash in\; \backslash tau\_\{\backslash omega\; X\}$, for every compact subset $C\; \backslash subseteq\; X$.

Let $X$ be a discrete topological space, and let $\backslash omega\; X$ be an Alexandroff one-point compactification of $X$. A Hausdorff space $P$ is polyadic if for some cardinal number $\backslash lambda$, there exists a continuous surjective function $f\; :\; \backslash omega\; X^\backslash lambda\; \backslash rightarrow\; P$, where $\backslash omega\; X^\backslash lambda$ is the product space obtained by multiplying $\backslash omega\; X$ with itself $\backslash lambda$ times.{{cite book|last=Turzański|first=Marian|title=Cantor Cubes: Chain Conditions|date=1996|page=19|isbn=978-8322607312|publisher=Wydawnictwo Uniwersytetu Śląskiego}}

Take the set of natural numbers $\backslash mathbb\{Z\}^+$ with the discrete topology. Its Alexandroff one-point compactification is $\backslash omega\; \backslash mathbb\{Z\}^+$. Choose $\backslash lambda\; =\; 1$ and define the homeomorphism $h\; :\; \backslash omega\; \backslash mathbb\{Z\}^+\; \backslash rightarrow\; \backslash left\; [\; 0,1\; \backslash right\; ]$ with the mapping

:$$

h(x) =

\begin{cases}

1/x, & \text{if }x\in\mathbb{Z}+ \\

0, & \text{if }x=\omega

\end{cases}

It follows from the definition that the image space $h[\backslash omega\backslash mathbb\; Z]\; =\; \backslash left\; \backslash \{0\; \backslash right\; \backslash \}\; \backslash cup\; \backslash left\; \backslash \{\; 1/n\; \backslash ,:\backslash ,\; n\; \backslash in\; \backslash mathbb\; N\; \backslash right\; \backslash \}$ is polyadic and compact directly from the definition of compactness, without using Heine-Borel.

Every dyadic space (a compact space which is a continuous image of a Cantor set{{cite book|page=[https://archive.org/details/moderngeneraltop0000naga/page/298 298]|title=Modern General Topology|chapter=Topics related to mappings|first=Jun-Iti|last=Nagata|isbn=978-0444876553|author-link=Jun-iti Nagata|date=1985-11-15|chapter-url=https://archive.org/details/moderngeneraltop0000naga/page/298}}) is a polyadic space.{{cite book|title=Abelian Groups, Module Theory, and Topology|first1=Dikran|last1=Dikranjan|first2=Luigi|last2=Salce|date=1998|page=339|publisher=CRC Press|isbn=9780824719371}}

Let X be a separable, compact space. If X is a metrizable space, then it is polyadic (the converse is also true).{{cite thesis|last=Al-Mahrouqi|first=Sharifa|date=2013|pages=8–13|title=Compact topological spaces inspired by combinatorial constructions|url=https://ueaeprints.uea.ac.uk/43361/|publisher=University of East Anglia}}

The cellularity $c(X)$ of a space $X$ is

c(X) = \sup \left \{ \vert B \vert : B \text{ is a disjoint collection of open sets of } X \right \}

The tightness $t(X)$ of a space $X$ is defined as follows: let $A\; \backslash subseteq\; X$, and $p\; \backslash in\; \backslash bar\{A\}$. Define

a(p, A) := \min \left \{ \vert B \vert : p \in \mathrm{cl}_X (B), B \subseteq A \right \}

t(p, X) := \sup \left \{ a(p, A) : A \subseteq X, p \in \mathrm{cl}_X(A) \right \}

Then $t(X)\; :=\; \backslash sup\; \backslash left\; \backslash \{\; t(p,\; X)\; :\; p\; \backslash in\; X\; \backslash right\; \backslash \}.${{cite journal|first=Murray|last=Bell|url=http://topology.auburn.edu/tp/reprints/v25/tp25105.pdf|title=Tightness in Polyadic Spaces|volume=25|year=2005|pages=2–74|journal=Topology Proceedings|publisher=Auburn University}}

The topological weight $w(X)$ of a polyadic space $X$ satisfies the equality $w(X)\; =\; c(X)\; \backslash cdot\; t(X)$.

Let $X$ be a polyadic space, and let $A\; \backslash subseteq\; X$. Then there exists a polyadic space $P\; \backslash subseteq\; X$ such that $A\; \backslash subseteq\; P$ and $c(P)\; \backslash le\; c(A)$.

Polyadic spaces are the smallest class of topological spaces that contain metric compact spaces and are closed under products and continuous images.{{cite arXiv|title=Universal Objects and Associations Between Classes of Banach Spaces and Classes of Compact Spaces|first=Piotr|last=Koszmider|eprint=1209.4294|date=2012|class=math.FA}} Every polyadic space $X$ of weight $\backslash leq\; 2^\backslash omega$ is a continuous image of $\backslash mathbb\{Z\}^+$.

A topological space $X$ has the Suslin property if there is no uncountable family of pairwise disjoint non-empty open subsets of $X$.{{cite web|url=http://www.ohio.edu/people/eisworth/research/Topology%20Group/Topology%20Comps/August%202005.pdf|title=Topology Comprehensive Exam|date=2005|access-date=2015-02-14|archive-date=2015-02-14 |archive-url=https://web.archive.org/web/20150214121934/http://www.ohio.edu/people/eisworth/research/Topology%20Group/Topology%20Comps/August%202005.pdf|publisher=Ohio University}} Suppose that $X$ has the Suslin property and is polyadic. Then $X$ is dyadic.{{cite journal|title=On generalizations of dyadic spaces|url=http://dml.cz/dmlcz/701809|first=Marian|last=Turzański|page=154|volume=30|issue=2|date=1989|issn=0001-7140|journal=Acta Universitatis Carolinae. Mathematica et Physica}}

Let $\backslash mathrm\{dis\}(X)$ be the least number of discrete sets needed to cover $X$, and let $\backslash Delta\; (X)$ denote the least cardinality of a non-empty open set in $X$. If $X$ is a polyadic space, then $\backslash mathrm\{dis\}(X)\; \backslash ge\; \backslash Delta\; (X)$.{{cite journal|journal=Commentationes Mathematicae Universitatis Carolinae|volume=50 |issue=3 |pages=463–475|title=A note on discrete sets|first=Santi|last=Spadaro|arxiv=0905.3588|date=2009-05-22}}

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 $CO(X)$ denote the clopen algebra of all clopen subsets of $X$. We define a Boolean space as a compact Hausdorff space whose basis is $CO(X)$. The element $G\; \backslash in\; CO(X)\text{'}$ such that $\backslash langle\backslash langle\; G\; \backslash rangle\backslash rangle\; =\; CO(X)$ is called the generating set for $CO(X)$. We say $G$ is a $(\backslash tau,\; \backslash kappa)$-disjoint collection if $G$ is the union of at most $\backslash tau$ subcollections $G\_\backslash alpha$, where for each $\backslash alpha$, $G\_\backslash alpha$ is a disjoint collection of cardinality at most $\backslash kappa$ It was proven by Petr Simon that $X$ is a Boolean space with the generating set $G$ of $CO(X)$ being $(\backslash tau,\; \backslash kappa)$-disjoint if and only if $X$ is homeomorphic to a closed subspace of $\backslash alpha\; \backslash kappa\; ^\; \backslash tau$. 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.{{cite web|url=http://www.utm.edu/staff/jschomme/topology/c/a/a/b/72.htm|first=Murray|last=Bell|title= A Ramsey Theorem for Polyadic Spaces|access-date=2015-02-14|date=1996-01-11|publisher=University of Tennessee at Martin}}

We define the compactness number of a space $X$, denoted by $\backslash mathrm\{cmpn\}(X)$, to be the least number $n$ such that $X$ 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 $\backslash mathcal\{S\}$ be a collection of sets and let $S$ be a set. We denote the set $\backslash left\backslash \{\backslash bigcap\; \backslash mathcal\{F\}\; :\; \backslash mathcal\{F\}\; \backslash text\{\; is\; a\; finite\; subset\; of\; \}\; \backslash mathcal\{S\}\backslash right\backslash \}$ by $\backslash mathcal\{S\}^\{\backslash widehat\{\backslash mathcal\{F\}\}\}$; all subsets of $S$ of size $n$ by $[S]^n$; and all subsets of size at most $n$ by $[S]^\{\backslash leq\; n\}$. If and $\backslash bigcap\; \backslash mathcal\{F\}\; \backslash ne\; \backslash empty$ for all $\backslash mathcal\{F\}\; \backslash in\; [\backslash mathcal\{S\}]^n$, then we say that $\backslash mathcal\{S\}$ is n-linked. If every n-linked subset of $\backslash mathcal\{S\}$ has a non-empty intersection, then we say that $\backslash mathcal\{S\}$ is n-ary. Note that if $\backslash mathcal\{S\}$ is n-ary, then so is $\backslash mathcal\{S\}^\{\backslash widehat\{\backslash mathcal\{F\}\}\}$, and therefore every space $X$ with $\backslash mathrm\{cmpn\}(X)\; \backslash leq\; n$ has a closed, n-ary subbase $\backslash mathcal\{S\}$ with $\backslash mathcal\{S\}\; =\; \backslash mathcal\{S\}^\{\backslash widehat\{\backslash mathcal\{F\}\}\}$. Note that a collection $\backslash mathcal\{S\}\; =\; \backslash mathcal\{S\}^\{\backslash widehat\{\backslash mathcal\{F\}\}\}$ of closed subsets of a compact space $X$ is a closed subbase if and only if for every closed $K$ in an open set $U$, there exists a finite $\backslash mathcal\{F\}$ such that $\backslash mathcal\{F\}\; \backslash subset\; \backslash mathcal\{S\}$ and $K\; \backslash subset\; \backslash bigcup\; \backslash mathcal\{F\}\; \backslash subset\; U$.{{cite journal|first=Murray|last=Bell|url=http://dml.cz/dmlcz/106376|date=1985|volume=26|number=2|pages=353–361|title=Polyadic spaces of arbitrary compactness numbers|journal=Commentationes Mathematicae Universitatis Carolinae|publisher=Charles University in Prague|access-date=2015-02-27}}

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

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

Proof: For $s\; \backslash in\; S$, define and $R\_s\; =\; \backslash \{\; F\; \backslash in\; s^+\; :\; |\; \backslash \{\; t\; \backslash in\; F\; :\; t\; >\; s\; \backslash \}\; |\; \backslash leq\; n\; -\; 1\; \backslash \}$. Set $\backslash mathcal\{R\}\; =\; \backslash bigcup\_\{\; s\; \backslash in\; S\}\; \backslash \{\; L\_s,\; R\_s,\; s^+\; \backslash \}$. For $A$, $B$ and $C$ such that $A\; \backslash cup\; B\; \backslash cup\; C\; \backslash ne\; \backslash empty$, let $\backslash mathcal\{F\}\; =\; \backslash \{\; L\_s\; :\; s\; \backslash in\; A\; \backslash \}\; \backslash cup\; \backslash \{\; R\_s\; :\; s\; \backslash in\; B\; \backslash \}\; \backslash cup\; \backslash \{\; s^-\; :\; s\; \backslash in\; C\; \backslash \}$ such that $\backslash mathcal\{F\}$ is an $n+1$-linked subset of $\backslash mathcal\{R\}$. Show that $A\; \backslash cup\; B\; \backslash in\; \backslash bigcap\; \backslash mathcal\{F\}$. $\backslash blacksquare$

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

Theorem (Lower bound on $\backslash mathrm\{cmpn\}\backslash left([S]^\{\backslash le\; n\}\backslash right)$) Let $n$ be such that . Then $[\backslash omega\_1]^\{\backslash leq\; 2n-1\}$ cannot be embedded as a neighbourhood retract in any space $K$ with $\backslash mathrm\{cmpn\}(K)\; \backslash leq\; n$.

From the two theorems above, it can be deduced that for $n$ such that , we have that $\backslash mathrm\{cmpn\}\backslash left([\backslash omega\_1]^\{\backslash leq\; 2n-1\}\backslash right)\; =\; n\; +\; 1\; =\; \backslash mathrm\{cmpn\}\backslash left([\backslash omega\_1]^\{\backslash leq\; 2n\}\backslash right)$.

Let $A$ be the Alexandroff one-point compactification of the discrete space $S$, so that $A\; =\; S\; \backslash cup\; \backslash \{\backslash infty\backslash \}$. We define the continuous surjection $g\; :\; A^n\; \backslash rightarrow\; [S]^\{\backslash leq\; n\}$ by $g((x\_1,\; ...,\; x\_n))\; =\; \backslash \{x\_1,\; \backslash ldots\; ,\; x\_n\backslash \}\; \backslash cap\; S$. It follows that $[S]^\{\backslash leq\; n\}$ is a polyadic space. Hence $[\backslash omega\_1]^\{\backslash leq\; 2n-1\}$ is a polyadic space with compactness number $\backslash mathrm\{cmpn\}\backslash left([\backslash omega\_1]^\{\backslash leq\; 2n-1\}\backslash right)\; =\; n+1$.

Centred spaces, AD-compact spaces{{cite journal|first=Grzegorz|last=Plebanek|date=1995-08-25|title=Compact spaces that result from adequate families of sets|volume=65|issue=3|pages=257–270|journal=Topology and Its Applications|publisher=Elsevier|doi=10.1016/0166-8641(95)00006-3|doi-access=free}} and ξ-adic spaces{{cite journal|pages=41–49|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm158/fm15814.pdf|title=On character and chain conditions in images of products|journal=Fundamenta Mathematicae|volume=158|issue=1|date=1998|publisher=Polish Academy of Sciences|first=Murray|last=Bell}} are generalisations of polyadic spaces.

Let $\backslash mathcal\{S\}$ be a collection of sets. We say that $\backslash mathcal\{S\}$ is centred if $\backslash bigcap\; \backslash mathcal\{F\}\; \backslash ne\; \backslash empty$ for all finite subsets $\backslash mathcal\{F\}\; \backslash subseteq\; \backslash mathcal\{S\}$.{{cite journal

| last = Bell | first = Murray G.

| doi = 10.4064/fm-125-1-47-58

| issue = 1

| journal = Fundamenta Mathematicae

| mr = 813988

| pages = 47–58

| title = Generalized dyadic spaces

| volume = 125

| year = 1985}} Define the Boolean space $Cen(\; \backslash mathcal\{S\}\; )\; =\; \backslash \{\; \backslash chi\_T\; :\; T\; \backslash text\{\; is\; a\; centred\; subcollection\; of\; \}\; \backslash mathcal\{S\}\; \backslash \}$, with the subspace topology from $2^\{\backslash mathcal\{S\}\}$. We say that a space $X$ is a centred space if there exists a collection $\backslash mathcal\{S\}$ such that $X$ is a continuous image of $Cen(\backslash mathcal\{S\})$.{{cite journal|url=http://dml.cz/dmlcz/127939|title=Function spaces on τ-Corson compacta and tightness of polyadic spaces|first=Murray|last=Bell|date=2004|journal=Czechoslovak Mathematical Journal|number=4|volume=54|pages=899–914|doi=10.1007/s10587-004-6439-z|s2cid=123078792}}

Centred spaces were introduced by Murray Bell in 2004.

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

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

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

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

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

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).{{cite journal|url=http://dml.cz/dmlcz/700712|last=Gerlits|first=János|title=On m-adic spaces|editor-first=Josef|editor-last=Novák|pages=147–148|journal=General Topology and Its Relations to Modern Analysis and Algebra, Proceedings of the Third Prague Topological Symposium|date=1971|publisher=Academia Publishing House of the Czechoslovak Academy of Science|location=Prague}}

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

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

Hyadic spaces were introduced by Eric van Douwen.{{cite journal|url=https://www.ams.org/journals/proc/1988-104-02/S0002-9939-1988-0962841-6/S0002-9939-1988-0962841-6.pdf|last=Bell|first=Murray|title=G_k subspaces of hyadic spaces|date=1988|journal=Proceedings of the American Mathematical Society|volume=104|issue=2|pages=635–640|publisher=American Mathematical Society|doi=10.2307/2047025|jstor=2047025|s2cid=201914041 }} They are defined as follows.

Let $X$ be a Hausdorff space. We denote by $H(X)$ the hyperspace of $X$. We define the subspace $J\_2\; (X)$ of $H(X)$ by $\backslash \{\; F\; \backslash in\; H(X)\; :\; |F|\; \backslash le\; 2\; \backslash \}$. A base of $H(X)$ is the family of all sets of the form $\backslash langle\; U\_0,\; \backslash dots\; ,\; U\_n\; \backslash rangle\; =\; \backslash \{\; F\; \backslash in\; H(X)\; :\; F\; \backslash subseteq\; U\_0\; \backslash cup\; \backslash dots\; \backslash cup\; U\_n,\; F\; \backslash cap\; U\_i\; \backslash ne\; \backslash empty\; \backslash text\{\; for\; \}\; 0\; \backslash le\; i\; \backslash le\; n\; \backslash \}$, where $n$ is any integer, and $U\_i$ are open in $X$. If $X$ is compact, then we say a Hausdorff space $Y$ is hyadic if there exists a continuous surjection from $H(X)$ to $Y$.{{cite journal|title=Mappings from hyperspaces and convergent sequences|last=van Douwen|first=Eric K.|author-link=Eric van Douwen|date=1990|volume=34|issue=1|pages=35–45|journal=Topology and Its Applications|publisher=Elsevier|doi=10.1016/0166-8641(90)90087-i|doi-access=free}}

Polyadic spaces are hyadic.{{cite journal|title=On cardinal invariants and metrizability of topological inverse Clifford semigroups|journal=Topology and Its Applications|last=Banakh|first=Taras|date=2003|volume=128|issue=1|page=38|publisher=Elsevier|doi=10.1016/S0166-8641(02)00083-4|doi-access=free}}

• Dyadic space
• Eberlein compactum
• Stone space
• Stone–Čech compactification
• Supercompact space

{{reflist|2}}

Category:Properties of topological spaces

Category:General topology