View source for 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|Suslin condition]] and, moreover, any regular [[Cardinal number|cardinal number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d0342201.png" /> is the [[Calibre|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d0342202.png" /> in a dyadic compactum are dyadic compacta. Any non-isolated point of a dyadic compactum is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d0342203.png" />-point. Moreover, if the character of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d0342204.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d0342205.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d0342206.png" /> contains an Aleksandrov compactum of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d0342207.png" /> the vertex of which coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d0342208.png" />. The weight of an infinite dyadic compactum is equal to the least upper bound of the characters of the points, while the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d0342209.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d03422010.png" /> is metrizable if one of the following conditions is met: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d03422011.png" /> satisfies the first axiom of countability; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d03422012.png" /> is a continuous image of an ordered compactum; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d03422013.png" /> is hereditarily normal; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d03422014.png" /> is hereditarily dyadic; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d03422015.png" /> is a Fréchet–Urysohn space; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d03422016.png" /> is hereditarily separable; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d03422017.png" /> is a quotient space of a metric space.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Engelking,   "General topology" , PWN  (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.A. Efimov,   "Dyadic bicompacta"  ''Trans. Moscow Math. Soc.'' , '''14'''  (1965)  pp. 229–267  ''Trudy Moskov. Mat. Obshch.'' , '''14'''  (1965)  pp. 211–247</TD></TR></table>



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

A point in a topological space is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034220/d03422019.png" />-point if it is the limit of a (non-trivial) converging sequence.