Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
topological group; space of ordinals; $C_p(X)$
topological group; space of ordinals; $C_p(X)$
Summary:
Let $\tau$ be an uncountable regular cardinal and $G$ a $T_1$ topological group. We prove the following statements: (1) If $\tau$ is homeomorphic to a closed subspace of $G$, $G$ is Abelian, and the order of every non-neutral element of $G$ is greater than $5$ then $\tau\times\tau$ embeds in $G$ as a closed subspace. (2) If $G$ is Abelian, algebraically generated by $\tau\subset G$, and the order of every element does not exceed $3$ then $\tau\times \tau$ is not embeddable in $G$. (3) There exists an Abelian topological group $H$ such that $\omega_1$ is homeomorphic to a closed subspace of $H$ and $\{t^2:t\in T\}$ is not closed in $H$ whenever $T\subset H$ is homeomorphic to $\omega_1$. Some other results are obtained.
References:
[ARH] Arhangelskii A.: Topological function spaces. Math. Appl., vol. 78, Kluwer Academic Publisher, Dordrecht, 1992. MR 1144519
[BUZ] Buzyakova R.Z.: Ordinals in topological groups. Fund. Math. 196 (2007), 127-138. DOI 10.4064/fm196-2-3 | MR 2342623 | Zbl 1133.54022
[C&R] Comfort W.W., Ross K.A.: Pseudocompactness and uniform continuity in topological groups. Pacific J. Math. 16 (1966), 483-496. DOI 10.2140/pjm.1966.16.483 | MR 0207886 | Zbl 0214.28502
[ENG] Engelking R.: General Topology. Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, revised ed., 1989. MR 1039321 | Zbl 0684.54001
[KUN] Kunen K.: Set Theory. Elsevier, 1980. MR 0597342 | Zbl 0960.03033
[PON] Pontryagin L.S.: Continuous Groups. Moscow; English translation: {Topological Groups}, Princeton University Press, Princeton, 1939. Zbl 0659.22001
Search
Partner of
