Difference between revisions of "Dyadic space"
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| − | A [[Tikhonov space]] for which there is a [[compactification]] which is a [[dyadic compactum]]. The class of dyadic spaces contains all separable [[metric space]]s and is closed with respect to [[Tikhonov product]]s. Dyadic spaces display many of the properties of dyadic compacta. For instance, all pseudo-compact groups are dyadic, but non-dyadic finally compact groups exist. A dyadic space satisfies the [[Suslin condition]], but not every regular cardinal number $n\geq\mathbb{N}_{0}$ is the [[calibre]] of a dyadic space. Any regular closed set in a dyadic space is dyadic, but not all closed $G_\delta$ sets are. A dyadic space which satisfies the first axiom of countability is metrizable, but non-metrizable hereditarily normal, hereditarily separable dyadic spaces also exist. | + | A [[Tikhonov space]] for which there is a [[compactification]] which is a [[dyadic compactum]]. The class of dyadic spaces contains all separable [[metric space]]s and is closed with respect to [[Tikhonov product]]s. Dyadic spaces display many of the properties of dyadic compacta. For instance, all pseudo-compact groups are dyadic, but non-dyadic finally compact groups exist. A dyadic space satisfies the [[Suslin condition]], but not every regular cardinal number $n\geq\mathbb{N}_{0}$ is the [[calibre]] of a dyadic space. Any [[regular closed set]] in a dyadic space is dyadic, but not all closed $G_\delta$ sets are. A dyadic space which satisfies the first axiom of countability is metrizable, but non-metrizable hereditarily normal, hereditarily separable dyadic spaces also exist. |
Latest revision as of 20:40, 2 January 2021
A Tikhonov space for which there is a compactification which is a dyadic compactum. The class of dyadic spaces contains all separable metric spaces and is closed with respect to Tikhonov products. Dyadic spaces display many of the properties of dyadic compacta. For instance, all pseudo-compact groups are dyadic, but non-dyadic finally compact groups exist. A dyadic space satisfies the Suslin condition, but not every regular cardinal number $n\geq\mathbb{N}_{0}$ is the calibre of a dyadic space. Any regular closed set in a dyadic space is dyadic, but not all closed $G_\delta$ sets are. A dyadic space which satisfies the first axiom of countability is metrizable, but non-metrizable hereditarily normal, hereditarily separable dyadic spaces also exist.
References
| [1] | V.I. Ponomarev, "On dyadic spaces" Fund. Math. , 52 : 3 (1963) pp. 351–354 (In Russian) |
| [2] | B.A. Efimov, "On dyadic spaces" Soviet Math. Dokl. , 4 (1963) pp. 1131–1134 Dokl. Akad. Nauk SSSR , 151 : 5 (1963) pp. 1021–1024 |
Comments
Finally compact is usually termed Lindelöf in the West, cf. Lindelöf space. For regular closed set see Dyadic compactum.
Dyadic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dyadic_space&oldid=38663