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 topological space.
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.
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.
For a non-compact, locally compact Hausdorff topological space , we define the Alexandroff one-point compactification as the topological space with the set , denoted , where , with the topology defined as follows:
- , for every compact subset .
Let be a discrete topological space, and let be an Alexandroff one-point compactification of . A Hausdorff space is polyadic if for some cardinal number , there exists a continuous surjective function , where is the product space obtained by multiplying with itself times.
Take the set of natural numbers with the discrete topology. Its Alexandroff one-point compactification is . Choose and define the homeomorphism with the mapping
It follows from the definition that the space is polyadic and compact directly from the definition of compactness, without using Heine-Borel.
Let X be a separable, compact space. If X is a metrizable space, then it is polyadic (the converse is also true).
The cellularity of a space is . The tightness of a space is defined as follows: let , and . We define , and define . Then The topological weight of a polyadic space satisfies the equality .
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 of weight is a continuous image of .
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.
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 denote the clopen algebra of all clopen subsets of . We define a Boolean space as a compact Hausdorff space whose basis is . The element such that is called the generating set for . We say is a -disjoint collection if is the union of at most subcollections , where for each , is a disjoint collection of cardinality at most It was proven by Petr Simon that is a Boolean space with the generating set of being -disjoint if and only if is homeomorphic to a closed subspace of . 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.
We define the compactness number of a space , denoted by , to be the least number such that 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 be a collection of sets and let be a set. We denote the set by ; all subsets of of size by ; and all subsets of size at most by . If and for all , then we say that is n-linked. If every n-linked subset of has a non-empty intersection, then we say that is n-ary. Note that if is n-ary, then so is , and therefore every space with has a closed, n-ary subbase with . Note that a collection of closed subsets of a compact space is a closed subbase if and only if for every closed in an open set , there exists a finite such that and .
Let be an infinite set and let by a number such that . We define the product topology on as follows: for , let , and let . Let be the collection . We take as a clopen subbase for our topology on . This topology is compact and Hausdorff. For and such that , we have that is a discrete subspace of , and hence that is a union of discrete subspaces.
Theorem (Upper bound on ): For each total order on , there is an -ary closed subbase of .
Proof: For , define and . Set . For , and such that , let such that is an -linked subset of . Show that .
For a topological space and a subspace , we say that a continuous function is a retraction if is the identity map on . We say that is a retract of . If there exists an open set such that , and is a retract of , then we say that is a neighbourhood retract of .
Theorem (Lower bound on ) Let be such that . Then cannot be embedded as a neighbourhood retract in any space with .
From the two theorems above, it can be deduced that for such that , we have that .
Let be the Alexandroff one-point compactification of the discrete space , so that . We define the continuous surjection by . It follows that is a polyadic space. Hence is a polyadic space with compactness number .
Let be a collection of sets. We say that is centred if for all finite subsets . Define the Boolean space , with the subspace topology from . We say that a space is a centred space if there exists a collection such that is a continuous image of .
Centred spaces were introduced by Murray Bell in 2004.
Let be a non-empty set, and consider a family of its subsets . We say that is an adequate family if:
- given , if every finite subset of is in , then .
We may treat as a topological space by considering it a subset of the Cantor cube , and in this case, we denote it .
Let be a compact space. If there exist a set and an adequate family , such that is the continuous image of , then we say that 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 be an infinite cardinal, and let be a topological space. We say that has the property if for any family of non-empty open subsets of , where , we can find a set and a point such that and for each neighbourhood of , we have that .
If is a ξ-adic space, then has the property for each infinite cardinal . 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. They are defined as follows.
Let be a Hausdorff space. We denote by the hyperspace of . We define the subspace of by . A base of is the family of all sets of the form , where is any integer, and are open in . If is compact, then we say a Hausdorff space is hyadic if there exists a continuous surjection from to .
- Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2003). "Dyadic compacta". Encyclopedia of General Topology. Elsevier Science. p. 193. ISBN 978-0444503558.
- Al-Mahrouqi, Sharifa (2013). Compact topological spaces inspired by combinatorial constructions (Thesis). University of East Anglia. pp. 8–13.
- Møller, Jesper M. (2014). "Topological spaces and continuous maps". General Topology. p. 58. ISBN 9781502795878.
- Tkachuk, Vladimir V. (2011). "Basic Notions of Topology and Function Spaces". A Cp-Theory Problem Book: Topological and Function Spaces. Springer Science+Business Media. p. 35. ISBN 9781441974426.
- Turzański, Marian (1996). Cantor Cubes: Chain Conditions. Wydawnictwo Uniwersytetu Śląskiego. p. 19. ISBN 978-8322607312.
- Nagata, Jun-Iti (1985-11-15). "Topics related to mappings". Modern General Topology. p. 298. ISBN 978-0444876553.
- Dikranjan, Dikran; Salce, Luigi (1998). Abelian Groups, Module Theory, and Topology. CRC Press. p. 339. ISBN 9780824719371.
- Bell, Murray (2005). "Tightness in Polyadic Spaces" (PDF). Topology Proceedings. Auburn University. 25: 2–74.
- Spadaro, Santi (2009-05-22). "A note on discrete sets". Commentationes Mathematicae Universitatis Carolinae. 50 (3): 463–475. arXiv:0905.3588.
- Koszmider, Piotr (2012). "Universal Objects and Associations Between Classes of Banach Spaces and Classes of Compact Spaces". arXiv:1209.4294 [math.FA].
- "Topology Comprehensive Exam" (PDF). Ohio University. 2005. Archived from the original (PDF) on 2015-02-14. Retrieved 2015-02-14.
- Turzański, Marian (1989). "On generalizations of dyadic spaces". Acta Universitatis Carolinae. Mathematica et Physica. 30 (2): 154. ISSN 0001-7140.
- Bell, Murray (1996-01-11). "A Ramsey Theorem for Polyadic Spaces". University of Tennessee at Martin. Retrieved 2015-02-14.
- Bell, Murray (1985). "Polyadic spaces of arbitrary compactness numbers". Commentationes Mathematicae Universitatis Carolinae. Charles University in Prague. 26 (2): 353–361. Retrieved 2015-02-27.
- Plebanek, Grzegorz (1995-08-25). "Compact spaces that result from adequate families of sets". Topology and its Applications. Elsevier. 65 (3): 257–270. doi:10.1016/0166-8641(95)00006-3.
- Bell, Murray (1998). "On character and chain conditions in images of products" (PDF). Fundamenta Mathematicae. Polish Academy of Sciences. 158 (1): 41–49.
- Bell, Murray. "Generalized dyadic spaces" (PDF): 47–58. Archived (PDF) from the original on 2011-06-08. Retrieved 2014-02-27. Cite journal requires
- Bell, Murray (2004). "Function spaces on τ-Corson compacta and tightness of polyadic spaces". Czechoslovak Mathematical Journal. 54 (4): 899–914. doi:10.1007/s10587-004-6439-z.
- Gerlits, János (1971). Novák, Josef (ed.). "On m-adic spaces". General Topology and Its Relations to Modern Analysis and Algebra, Proceedings of the Third Prague Topological Symposium. Prague: Academia Publishing House of the Czechoslovak Academy of Science: 147–148.
- Bell, Murray (1988). "Gₖ subspaces of hyadic spaces" (PDF). Proceedings of the American Mathematical Society. American Mathematical Society. 104 (2): 635–640. doi:10.2307/2047025. JSTOR 2047025.
- van Douwen, Eric K. (1990). "Mappings from hyperspaces and convergent sequences". Topology and Its Applications. Elsevier. 34 (1): 35–45. doi:10.1016/0166-8641(90)90087-i.
- Banakh, Taras (2003). "On cardinal invariants and metrizability of topological inverse Clifford semigroups". Topology and Its Applications. Elsevier. 128 (1): 38. doi:10.1016/S0166-8641(02)00083-4.