Polyadic space

In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.
History.
Polyadic spaces were first studied by S. Mrówka in 1970 as a generalisation of dyadic spaces. 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.
Background.
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.
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 formula_1, we define the Alexandroff one-point compactification as the topological space with the set formula_2, denoted formula_3, where formula_4, with the topology formula_5 defined as follows:
Definition.
Let formula_9 be a discrete topological space, and let formula_3 be an Alexandroff one-point compactification of formula_9. A Hausdorff space formula_12 is polyadic if for some cardinal number formula_13, there exists a continuous surjective function formula_14, where formula_15 is the product space obtained by multiplying formula_3 with itself formula_13 times.
Examples.
Take the set of natural numbers formula_18 with the discrete topology. Its Alexandroff one-point compactification is formula_19. Choose formula_20 and define the homeomorphism formula_21 with the mapping
It follows from the definition that the space formula_23 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) is a polyadic space.
Let "X" be a separable, compact space. If "X" is a metrizable space, then it is polyadic (the converse is also true).
Properties.
The cellularity formula_24 of a space formula_9 is formula_26. The tightness formula_27 of a space formula_9 is defined as follows: let formula_29, and formula_30. We define formula_31, and define formula_32. Then formula_33 The topological weight formula_34 of a polyadic space formula_9 satisfies the equality formula_36.
Let formula_9 be a polyadic space, and let formula_29. Then there exists a polyadic space formula_39 such that formula_40 and formula_41.
Polyadic spaces are the smallest class of topological spaces that contain metric compact spaces and are closed under products and continuous images. Every polyadic space formula_9 of weight formula_43 is a continuous image of formula_44.
A topological space "X" has the Suslin property if there is no uncountable family of pairwise disjoint non-empty open subsets of X. Suppose that "X" has the Suslin property and "X" is polyadic. Then "X" is dyadic.
Let formula_45 be the least number of discrete sets needed to cover formula_9, and let formula_47 denote the least cardinality of a non-empty open set in formula_9. If formula_9 is a polyadic space, then formula_50.
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 formula_51 denote the clopen algebra of all clopen subsets of formula_9. We define a Boolean space as a compact Hausdorff space whose basis is formula_51. The element formula_54 such that formula_55 is called the generating set for formula_51. We say formula_57 is a formula_58-disjoint collection if formula_57 is the union of at most formula_60 subcollections formula_61, where for each formula_62, formula_61 is a disjoint collection of cardinality at most formula_64 It was proven by Petr Simon that formula_9 is a Boolean space with the generating set formula_57 of formula_51 being formula_58-disjoint if and only if formula_9 is homeomorphic to a closed subspace of formula_70. 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.
Compactness.
We define the compactness number of a space formula_9, denoted by formula_72, to be the least number formula_73 such that formula_9 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 formula_75 be a collection of sets and let formula_76 be a set. We denote the set formula_77 by formula_78; all subsets of formula_76 of size formula_73 by formula_81; and all subsets of size at most formula_73 by formula_83. If formula_84 and formula_85 for all formula_86, then we say that formula_75 is n-linked. If every n-linked subset of formula_75 has a non-empty intersection, then we say that formula_75 is n-ary. Note that if formula_75 is n-ary, then so is formula_78, and therefore every space formula_9 with formula_93 has a closed, n-ary subbase formula_75 with formula_95. Note that a collection formula_95 of closed subsets of a compact space formula_9 is a closed subbase if and only if for every closed formula_98 in an open set formula_99, there exists a finite formula_100 such that formula_101 and formula_102.
Let formula_76 be an infinite set and let formula_73 by a number such that formula_105. We define the product topology on formula_106 as follows: for formula_107, let formula_108, and let formula_109. Let formula_75 be the collection formula_111. We take formula_75 as a clopen subbase for our topology on formula_106. This topology is compact and Hausdorff. For formula_114 and formula_73 such that formula_116, we have that formula_117 is a discrete subspace of formula_106, and hence that formula_106 is a union of formula_120 discrete subspaces.
Theorem (Upper bound on formula_121): For each total order formula_122 on formula_76, there is an formula_120-ary closed subbase formula_125 of formula_126.
Proof: For formula_107, define formula_128 and formula_129. Set formula_130. For formula_131, formula_132 and formula_133 such that formula_134, let formula_135 such that formula_100 is an formula_120-linked subset of formula_125. Show that formula_139. formula_140
For a topological space formula_9 and a subspace formula_142, we say that a continuous function formula_143 is a retraction if formula_144 is the identity map on formula_131. We say that formula_131 is a retract of formula_9. If there exists an open set formula_99 such that formula_149, and formula_131 is a retract of formula_99, then we say that formula_131 is a neighbourhood retract of formula_9.
Theorem (Lower bound on formula_121) Let formula_73 be such that formula_84. Then formula_157 cannot be embedded as a neighbourhood retract in any space formula_98 with formula_159.
From the two theorems above, it can be deduced that for formula_73 such that formula_161, we have that formula_162.
Let formula_131 be the Alexandroff one-point compactification of the discrete space formula_76, so that formula_165. We define the continuous surjection formula_166 by formula_167. It follows that formula_106 is a polyadic space. Hence formula_157 is a polyadic space with compactness number formula_170.
Generalisations.
Centred spaces, AD-compact spaces and ξ-adic spaces are generalisations of polyadic spaces.
Centred space.
Let formula_75 be a collection of sets. We say that formula_75 is centred if formula_85 for all finite subsets formula_174. Define the Boolean space formula_175, with the subspace topology from formula_176. We say that a space formula_9 is a centred space if there exists a collection formula_75 such that formula_9 is a continuous image of formula_180.
Centred spaces were introduced by Murray Bell in 2004.
AD-compact space.
Let formula_9 be a non-empty set, and consider a family of its subsets formula_182. We say that formula_183 is an adequate family if:
We may treat formula_183 as a topological space by considering it a subset of the Cantor cube formula_190, and in this case, we denote it formula_191.
Let formula_98 be a compact space. If there exist a set formula_9 and an adequate family formula_182, such that formula_98 is the continuous image of formula_191, then we say that formula_98 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.
ξ-adic space.
Let formula_64 and formula_60 be cardinals, and let formula_9 be a Hausdorff space. If there exists a continuous surjection from formula_201 to formula_9, then formula_9 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).
Let formula_204 be an infinite cardinal, and let formula_9 be a topological space. We say that formula_9 has the property formula_207 if for any family formula_208 of non-empty open subsets of formula_9, where formula_210, we can find a set formula_211 and a point formula_212 such that formula_213 and for each neighbourhood formula_214 of formula_215, we have that formula_216.
If formula_9 is a ξ-adic space, then formula_9 has the property formula_207 for each infinite cardinal formula_204. It follows from this result that no infinite ξ-adic Hausdorff space can be an extremally disconnected space.
Hyadic space.
Hyadic spaces were introduced by Eric van Douwen. They are defined as follows.
Let formula_9 be a Hausdorff space. We denote by formula_222 the hyperspace of formula_9. We define the subspace formula_224 of formula_222 by formula_226. A base of formula_222 is the family of all sets of the form formula_228, where formula_73 is any integer, and formula_230 are open in formula_9. If formula_9 is compact, then we say a Hausdorff space formula_233 is hyadic if there exists a continuous surjection from formula_222 to formula_233.
Polyadic spaces are hyadic.