# Polyadic space

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{{Short description|Type of topological space}}
In [mathematics](/source/mathematics), a '''polyadic space''' is a [topological space](/source/topological_space) that is the [image](/source/image_(mathematics)) under a [continuous function](/source/continuous_function_(topology)) of a [topological power](/source/product_topology) of an [Alexandroff one-point compactification](/source/Alexandroff_one-point_compactification) of a [discrete space](/source/discrete_space).

== History ==
Polyadic spaces were first studied by S. Mrówka in 1970 as a generalisation of [dyadic space](/source/dyadic_space)s.<ref name="Hart2003">{{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](/source/Elsevier_Science)|isbn=978-0444503558|date=2003}}</ref> The theory was developed further by R. H. Marty, János Gerlits and Murray G. Bell,<ref name="Mahrouqi2013"/> the latter of whom introduced the concept of the more general [centred space](/source/centred_space)s.<ref name="Hart2003"/>

== Background ==
A subset ''K'' of a topological space ''X'' is said to be [compact](/source/compact_set) if every open [cover](/source/cover_(topology)) 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](/source/locally_compact_space) if it is locally compact at every point in the space.<ref name="Moller2014">{{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 }}</ref>

A proper subset ''A'' ⊂ ''X'' is said to be [dense](/source/dense_set) if the [closure](/source/closure_(topology)) ''Ā'' = ''X''. A space whose set has a countable, dense subset is called a [separable space](/source/separable_space).

For a non-compact, locally compact Hausdorff topological space <math>(X, \tau_X)</math>, we define the Alexandroff one-point compactification as the topological space with the set <math>\left \{ \omega \right \} \cup X</math>, denoted <math>\omega X</math>, where <math>\omega \notin X</math>, with the topology <math>\tau_{\omega X}</math> defined as follows:<ref name="Mahrouqi2013"/><ref name="Tkachuk2011">{{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](/source/Springer_Science%2BBusiness_Media)}}</ref>
* <math>\tau_X \subseteq \tau_{\omega X}</math>
* <math>X \setminus C \cup \left \{ \omega \right \} \in \tau_{\omega X}</math>, for every compact subset <math>C \subseteq X</math>.

== Definition ==
Let <math>X</math> be a discrete topological space, and let <math>\omega X</math> be an Alexandroff one-point compactification of <math>X</math>. A Hausdorff space <math>P</math> is polyadic if for some [cardinal number](/source/cardinal_number) <math>\lambda</math>, there exists a continuous surjective function <math>f : \omega X^\lambda \rightarrow P</math>, where <math>\omega X^\lambda</math> is the product space obtained by multiplying <math>\omega X</math> with itself <math>\lambda</math> times.<ref name="Turzanski1996">{{cite book|last=Turzański|first=Marian|title=Cantor Cubes: Chain Conditions|date=1996|page=19|isbn=978-8322607312|publisher=[Wydawnictwo Uniwersytetu Śląskiego](/source/Wydawnictwo_Uniwersytetu_%C5%9Al%C4%85skiego)}}</ref>

== Examples ==
Take the set of natural numbers <math>\mathbb{Z}^+</math> with the discrete topology. Its Alexandroff one-point compactification is <math>\omega \mathbb{Z}^+</math>. Choose <math>\lambda = 1</math> and define the homeomorphism <math>h : \omega \mathbb{Z}^+ \rightarrow \left [ 0,1 \right ]</math> with the mapping
:<math>
h(x) =
\begin{cases}
1/x, & \text{if }x\in\mathbb{Z}+ \\
0, & \text{if }x=\omega
\end{cases}
</math>
It follows from the definition that the image space <math>h[\omega\mathbb Z] = \left \{0 \right \} \cup \left \{ 1/n \,:\, n \in \mathbb N \right \}</math> 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<ref name="Nagata1985">{{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|publisher=Elsevier Science |chapter-url=https://archive.org/details/moderngeneraltop0000naga/page/298}}</ref>) is a polyadic space.<ref>{{cite book|title=Abelian Groups, Module Theory, and Topology|first1=Dikran|last1=Dikranjan|first2=Luigi|last2=Salce|date=1998|page=339|publisher=[CRC Press](/source/CRC_Press)|isbn=9780824719371}}</ref>

Let ''X'' be a separable, compact space. If ''X'' is a [metrizable space](/source/metrizable_space), then it is polyadic (the converse is also true).<ref name="Mahrouqi2013">{{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](/source/University_of_East_Anglia)}}</ref>

== Properties ==
The cellularity <math>c(X)</math> of a space <math>X</math> is
<math display="block">
c(X) = \sup \left \{ \vert B \vert : B \text{ is a disjoint collection of open sets of } X \right \}
</math>

The tightness <math>t(X)</math> of a space <math>X</math> is defined as follows: let <math>A \subseteq X</math>, and <math>p \in \bar{A}</math>. Define
<math display="block">
a(p, A) := \min \left \{ \vert  B \vert : p \in \mathrm{cl}_X (B), B \subseteq A \right \}
</math>
<math display="block">
t(p, X) := \sup \left \{ a(p, A) : A \subseteq X, p \in \mathrm{cl}_X(A) \right \}
</math>
Then <math>t(X) := \sup \left \{ t(p, X) : p \in X \right \}.</math><ref name="Bell2005">{{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](/source/Auburn_University)}}</ref>

The [topological weight](/source/Base_(topology)) <math>w(X)</math> of a polyadic space <math>X</math> satisfies the equality <math>w(X) = c(X) \cdot t(X)</math>.<ref name="Spadaro2009"/>

Let <math>X</math> be a polyadic space, and let <math>A \subseteq X</math>. Then there exists a polyadic space <math>P \subseteq X</math> such that <math>A \subseteq P</math> and <math>c(P) \le c(A)</math>.<ref name="Spadaro2009"/>

Polyadic spaces are the smallest class of topological spaces that contain metric compact spaces and are closed under products and continuous images.<ref name="Piotr2012">{{cite arXiv|title=Universal Objects and Associations Between Classes of Banach Spaces and Classes of Compact Spaces|first=Piotr|last=Koszmider|eprint=1209.4294<!--|page=17-->|date=2012|class=math.FA}}</ref> Every polyadic space <math>X</math> of weight <math>\leq 2^\omega</math> is a continuous image of <math>\mathbb{Z}^+</math>.<ref name="Piotr2012"/>

A topological space <math>X</math> has the [Suslin property](/source/Suslin_property) if there is no uncountable family of pairwise disjoint non-empty open subsets of <math>X</math>.<ref name="TopologyExam2005">{{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](/source/Ohio_University)}}</ref> Suppose that <math>X</math> has the Suslin property and is polyadic. Then <math>X</math> is dyadic.<ref name="Turzanski1989">{{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}}</ref>

Let <math>\mathrm{dis}(X)</math> be the least number of discrete sets needed to cover <math>X</math>, and let <math>\Delta (X)</math> denote the least cardinality of a non-empty [open set](/source/open_set) in <math>X</math>. If <math>X</math> is a polyadic space, then <math>\mathrm{dis}(X) \ge \Delta (X)</math>.<ref name="Spadaro2009">{{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}}</ref>

=== Ramsey's theorem ===
There is an analogue of [Ramsey's theorem](/source/Ramsey's_theorem) from combinatorics for polyadic spaces. For this, we describe the relationship between [Boolean space](/source/Boolean_space)s and polyadic spaces. Let <math>CO(X)</math> denote the [clopen](/source/clopen) algebra of all clopen subsets of <math>X</math>. We define a Boolean space as a compact [Hausdorff space](/source/Hausdorff_space) whose basis is <math>CO(X)</math>. The element <math>G \in CO(X)'</math> such that <math>\langle\langle G \rangle\rangle = CO(X)</math> is called the generating set for <math>CO(X)</math>. We say <math>G</math> is a <math>(\tau, \kappa) </math>-disjoint collection if <math>G</math> is the union of at most <math>\tau</math> subcollections <math>G_\alpha</math>, where for each <math>\alpha</math>, <math>G_\alpha</math> is a disjoint collection of cardinality at most <math>\kappa</math> It was proven by Petr Simon that <math>X</math> is a Boolean space with the generating set <math>G</math> of <math>CO(X)</math> being <math>(\tau, \kappa) </math>-disjoint if and only if <math>X</math> is homeomorphic to a closed subspace of <math>\alpha \kappa ^ \tau</math>.<ref name="Bell2005"/> 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.<ref name="Bell2006">{{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](/source/University_of_Tennessee_at_Martin)}}</ref>

=== Compactness ===
We define the [compactness number](/source/compactness_number) of a space <math>X</math>, denoted by <math>\mathrm{cmpn}(X)</math>, to be the least number <math>n</math> such that <math>X</math> has an n-ary closed [subbase](/source/subbase). We can construct polyadic spaces with arbitrary compactness number. We will demonstrate this using two theorems proven by Murray Bell in 1985. Let <math>\mathcal{S}</math> be a collection of sets and let <math>S</math> be a set. We denote the set <math>\left\{\bigcap \mathcal{F} : \mathcal{F} \text{ is a finite subset of } \mathcal{S}\right\}</math> by <math>\mathcal{S}^{\widehat{\mathcal{F}}}</math>; all subsets of <math>S</math> of size <math>n</math> by <math>[S]^n</math>; and all subsets of size at most <math>n</math> by <math>[S]^{\leq n}</math>. If <math>2 \leq n < \omega</math> and <math>\bigcap \mathcal{F} \ne \empty</math> for all <math>\mathcal{F} \in [\mathcal{S}]^n</math>, then we say that <math>\mathcal{S}</math> is n-linked. If every n-linked subset of <math>\mathcal{S}</math> has a non-empty intersection, then we say that <math>\mathcal{S}</math> is n-ary. Note that if <math>\mathcal{S}</math> is n-ary, then so is <math>\mathcal{S}^{\widehat{\mathcal{F}}}</math>, and therefore every space <math>X</math> with <math>\mathrm{cmpn}(X) \leq n</math> has a closed, n-ary subbase <math>\mathcal{S}</math> with <math>\mathcal{S} = \mathcal{S}^{\widehat{\mathcal{F}}}</math>. Note that a collection <math>\mathcal{S} = \mathcal{S}^{\widehat{\mathcal{F}}}</math> of closed subsets of a compact space <math>X</math> is a closed subbase if and only if for every closed <math>K</math> in an open set <math>U</math>, there exists a finite <math>\mathcal{F}</math> such that <math>\mathcal{F} \subset \mathcal{S}</math> and <math>K \subset \bigcup \mathcal{F} \subset U</math>.<ref name="Bell1985">{{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](/source/Commentationes_Mathematicae_Universitatis_Carolinae)|publisher=[Charles University in Prague](/source/Charles_University_in_Prague)|access-date=2015-02-27}}</ref>

Let <math>S</math> be an infinite set and let <math>n</math> by a number such that <math>1 \leq n < \omega</math>. We define the [product topology](/source/product_topology) on <math>[S]^{\leq n}</math> as follows: for <math>s \in S</math>, let <math>s^- = \{F \in [S]^{\leq n} : s \in F\}</math>, and let <math>s^+ = \{F \in [S]^{\leq n} : s \notin F\}</math>. Let <math>\mathcal{S}</math> be the collection <math>\mathcal{S} = \bigcup_{s \in S} \{s^+, s^-\}</math>. We take <math>\mathcal{S}</math> as a clopen subbase for our topology on <math>[S]^{\leq n}</math>. This topology is compact and Hausdorff. For <math>k</math> and <math>n</math> such that <math>0 \le k \leq n</math>, we have that <math>[S]^k</math> is a discrete subspace of <math>[S]^{\leq n}</math>, and hence that <math>[S]^{\leq n}</math> is a union of <math>n+1</math> discrete subspaces.<ref name="Bell1985"/>

'''Theorem''' (Upper bound on <math>\mathrm{cmpn}\left([S]^{\leq n}\right)</math>): For each [total order](/source/total_order) <math><</math> on <math>S</math>, there is an <math>n+1</math>-ary closed subbase <math>\mathcal{R}</math> of <math>[S]^{\leq 2n}</math>.

'''Proof''': For <math>s \in S</math>, define <math>L_s = \{ F \in s^+ : | \{ t \in F : t < s \} | \leq n - 1 \}</math> and <math>R_s = \{ F \in s^+ : | \{ t \in F : t > s \} | \leq n - 1 \}</math>. Set <math>\mathcal{R} = \bigcup_{ s \in S} \{ L_s, R_s, s^+ \}</math>. For <math>A</math>, <math>B</math> and <math>C</math> such that <math>A \cup B \cup C \ne \empty</math>, let <math>\mathcal{F} = \{ L_s : s \in A \} \cup \{ R_s : s \in B \} \cup \{ s^- : s \in C \}</math> such that <math>\mathcal{F}</math> is an <math>n+1</math>-linked subset of <math>\mathcal{R}</math>. Show that <math>A \cup B \in \bigcap \mathcal{F}</math>. <math>\blacksquare</math>

For a topological space <math>X</math> and a subspace <math>A \in X</math>, we say that a continuous function <math>r : X \rightarrow A</math> is a [retraction](/source/deformation_retract) if <math>r|_A</math> is the identity map on <math>A</math>. We say that <math>A</math> is a retract of <math>X</math>. If there exists an open set <math>U</math> such that <math>A \subset U \subset X</math>, and <math>A</math> is a retract of <math>U</math>, then we say that <math>A</math> is a neighbourhood retract of <math>X</math>.

'''Theorem''' (Lower bound on <math>\mathrm{cmpn}\left([S]^{\le n}\right)</math>) Let <math>n</math> be such that <math>2 \leq n < \omega</math>. Then <math>[\omega_1]^{\leq 2n-1}</math> cannot be embedded as a neighbourhood retract in any space <math>K</math> with <math>\mathrm{cmpn}(K) \leq n</math>.

From the two theorems above, it can be deduced that for <math>n</math> such that <math>1 \leq n < \omega</math>, we have that <math>\mathrm{cmpn}\left([\omega_1]^{\leq 2n-1}\right) = n + 1 = \mathrm{cmpn}\left([\omega_1]^{\leq 2n}\right)</math>.

Let <math>A</math> be the Alexandroff one-point compactification of the discrete space <math>S</math>, so that <math>A = S \cup \{\infty\}</math>. We define the continuous surjection <math>g : A^n \rightarrow [S]^{\leq n}</math> by <math>g((x_1, ..., x_n)) = \{x_1, \ldots , x_n\} \cap S</math>. It follows that <math>[S]^{\leq n}</math> is a polyadic space. Hence <math>[\omega_1]^{\leq 2n-1}</math> is a polyadic space with compactness number <math>\mathrm{cmpn}\left([\omega_1]^{\leq 2n-1}\right) = n+1</math>.<ref name="Bell1985"/>

== Generalisations ==
Centred spaces, AD-compact spaces<ref name="Plebanek1995">{{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](/source/Topology_and_Its_Applications)|publisher=Elsevier|doi=10.1016/0166-8641(95)00006-3|doi-access=free}}</ref> and ξ-adic spaces<ref name="Bell1998">{{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](/source/Fundamenta_Mathematicae)|volume=158|issue=1|date=1998|publisher=[Polish Academy of Sciences](/source/Polish_Academy_of_Sciences)|first=Murray|last=Bell}}</ref> are generalisations of polyadic spaces.

=== Centred space ===
Let <math>\mathcal{S}</math> be a collection of sets. We say that <math>\mathcal{S}</math> is centred if <math>\bigcap \mathcal{F} \ne \empty</math> for all finite subsets <math>\mathcal{F} \subseteq \mathcal{S}</math>.<ref name="Bell1985b">{{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}}</ref> Define the Boolean space <math>Cen( \mathcal{S} ) = \{ \chi_T : T \text{ is a centred subcollection of } \mathcal{S} \}</math>, with the subspace topology from <math>2^{\mathcal{S}}</math>. We say that a space <math>X</math> is a centred space if there exists a collection <math>\mathcal{S}</math> such that <math>X</math> is a continuous image of <math>Cen(\mathcal{S})</math>.<ref name="Bell2004">{{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](/source/Czechoslovak_Mathematical_Journal)|number=4|volume=54|pages=899–914|doi=10.1007/s10587-004-6439-z|s2cid=123078792}}</ref>

Centred spaces were introduced by Murray Bell in 2004.

=== AD-compact space ===
Let <math>X</math> be a non-empty set, and consider a family of its subsets <math>\mathcal{A} \subseteq \mathcal{P} (X)</math>. We say that <math>\mathcal{A}</math> is an adequate family if:
* <math>A \in \mathcal{A} \land B \subseteq \mathcal{A} \Rightarrow B \in \mathcal{A}</math>
* given <math>A \subseteq X</math>, if every finite subset of <math>A</math> is in <math>\mathcal{A}</math>, then <math>A \in \mathcal{A}</math>.

We may treat <math>\mathcal{A}</math> as a topological space by considering it a subset of the [Cantor cube](/source/Cantor_cube) <math>D^X</math>, and in this case, we denote it <math>K(\mathcal{A})</math>.

Let <math>K</math> be a compact space. If there exist a set <math>X</math> and an adequate family <math>\mathcal{A} \subseteq \mathcal{P} (X)</math>, such that <math>K</math> is the continuous image of <math>K(\mathcal{A})</math>, then we say that <math>K</math> 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](/source/disjoint_union_(topology)). 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.<ref name="Plebanek1995"/>

=== ξ-adic space ===
Let <math>\kappa</math> and <math>\tau</math> be cardinals, and let <math>X</math> be a Hausdorff space. If there exists a continuous surjection from <math>(\kappa + 1)^\tau</math> to <math>X</math>, then <math>X</math> is said to be a ξ-adic space.<ref name="Bell1998"/>

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).<ref name="Gerlits1972">{{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](/source/Prague)}}</ref>

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

If <math>X</math> is a ξ-adic space, then <math>X</math> has the property <math>\mathbf{B} ( \mathfrak{n} )</math> for each infinite cardinal <math>\mathfrak{n}</math>. It follows from this result that no infinite ξ-adic Hausdorff space can be an [extremally disconnected space](/source/extremally_disconnected_space).<ref name="Gerlits1972"/>

=== Hyadic space ===
Hyadic spaces were introduced by [Eric van Douwen](/source/Eric_van_Douwen).<ref name="Bell1988">{{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<sub>k</sub> subspaces of hyadic spaces|date=1988|journal=[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society)|volume=104|issue=2|pages=635–640|publisher=[American Mathematical Society](/source/American_Mathematical_Society)|doi=10.2307/2047025|jstor=2047025|s2cid=201914041 }}</ref> They are defined as follows.

Let <math>X</math> be a Hausdorff space. We denote by <math>H(X)</math> the hyperspace of <math>X</math>. We define the subspace <math>J_2 (X)</math> of <math>H(X)</math> by <math>\{ F \in H(X) : |F| \le 2 \}</math>. A base of <math>H(X)</math> is the family of all sets of the form <math>\langle U_0, \dots , U_n \rangle = \{ F \in H(X) : F \subseteq U_0 \cup \dots \cup U_n, F \cap U_i \ne \empty \text{ for } 0 \le i \le n \}</math>, where <math>n</math> is any integer, and <math>U_i</math> are open in <math>X</math>. If <math>X</math> is compact, then we say a Hausdorff space <math>Y</math> is hyadic if there exists a continuous surjection from <math>H(X)</math> to <math>Y</math>.<ref name="VanDouwen1990">{{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}}</ref>

Polyadic spaces are hyadic.<ref>{{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}}</ref>

== See also ==
* [Dyadic space](/source/Dyadic_space)
* [Eberlein compactum](/source/Eberlein_compactum)
* [Stone space](/source/Stone_space)
* [Stone–Čech compactification](/source/Stone%E2%80%93%C4%8Cech_compactification)
* [Supercompact space](/source/Supercompact_space)

== References ==
{{reflist|2}}

Category:Properties of topological spaces
Category:General topology

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Adapted from the Wikipedia article [Polyadic space](https://en.wikipedia.org/wiki/Polyadic_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Polyadic_space?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.