Dyadic compactum

A compactum which is a continuous image of a generalized Cantor discontinuum. Dyadic compacta were introduced by P.S. Aleksandrov, who made a natural attempt to extend the theorem asserting that any metric compactum is a continuous image of a Cantor set to arbitrary compacta. The class of dyadic compacta is the smallest class of compacta containing all metric compacta and which is closed with respect to the Tikhonov product and continuous mappings. Properties of dyadic compacta are: Any compact topological group is dyadic. Dyadic compacta satisfy the Suslin condition and, moreover, any regular cardinal number \(\mathfrak m\geq\aleph_0\) is the calibre of a dyadic compactum. Hence it follows that non-dyadic compacta exist. These include, for example, all Aleksandrov compacta of uncountable cardinality (a one-point compactification of an infinite discrete space). All regular closed sets and all closed sets of type \(G_\delta\) in a dyadic compactum are dyadic compacta. Any non-isolated point of a dyadic compactum is a \(\kappa\)-point. Moreover, if the character of the point \(x\in X\) is \(\mathfrak m\geq\aleph_0\), then \(X\) contains an Aleksandrov compactum of cardinality \(\mathfrak m\) the vertex of which coincides with \(x\). The weight of an infinite dyadic compactum is equal to the least upper bound of the characters of the points, while the \(\pi\)-weight of a dyadic compactum is equal to its weight. Any extremally-disconnected dyadic compactum is finite. There exist various criteria for metrizability of dyadic compacta. In particular, a dyadic compactum \(X\) is metrizable if one of the following conditions is met: \(X\) satisfies the first axiom of countability; \(X\) is a continuous image of an ordered compactum; \(X\) is hereditarily normal; \(X\) is hereditarily dyadic; \(X\) is a Fréchet–Urysohn space; \(X\) is hereditarily separable; \(X\) is a quotient space of a metric space.

References

[1]	R. Engelking, "General topology" , PWN (1977)
[2]	J.L. Kelley, "General topology" , Springer (1975)
[3]	B.A. Efimov, "Dyadic bicompacta" Trans. Moscow Math. Soc. , 14 (1965) pp. 229–267 Trudy Moskov. Mat. Obshch. , 14 (1965) pp. 211–247

Comments

A regular closed set (sometimes called canonically closed set or canonical set) is the closure of an open set.

A point in a topological space is called a \(\kappa\)-point if it is the limit of a (non-trivial) converging sequence.