[1] The theory was developed further by R. H. Marty, János Gerlits and Murray G. Bell,[2] the latter of whom introduced the concept of the more general centred spaces.
[1] A subset K of a topological space X is said to be compact if every open cover of K contains a finite subcover.
[3] A proper subset A ⊂ X is said to be dense if the closure Ā = X.
For a non-compact, locally compact Hausdorff topological space
, we define the Alexandroff one-point compactification as the topological space with the set
is the product space obtained by multiplying
with the mapping It follows from the definition that the image space
If X is a metrizable space, then it is polyadic (the converse is also true).
is a disjoint collection of open sets of
{\displaystyle c(X)=\sup \left\{\vert B\vert :B{\text{ is a disjoint collection of open sets of }}X\right\}}
has the Suslin property if there is no uncountable family of pairwise disjoint non-empty open subsets of
be the least number of discrete sets needed to cover
denote the least cardinality of a non-empty open set in
[9] There is an analogue of Ramsey's theorem from combinatorics for polyadic spaces.
is a Boolean space with the generating set
[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] We define the compactness number of a space
We can construct polyadic spaces with arbitrary compactness number.
We will demonstrate this using two theorems proven by Murray Bell in 1985.
of closed subsets of a compact space
cannot be embedded as a neighbourhood retract in any space
be the Alexandroff one-point compactification of the discrete space
is a polyadic space with compactness number
is a centred space if there exists a collection
[18] Centred spaces were introduced by Murray Bell in 2004.
as a topological space by considering it a subset of the Cantor cube
AD-compact spaces were introduced by Grzegorz Plebanek.
He proved that they are closed under arbitrary products and Alexandroff compactifications of disjoint unions.
The converse is not true, as there are AD-compact spaces that are not polyadic.
[19] Hyadic spaces were introduced by Eric van Douwen.