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The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, $m (α) \leq 2^{α}$ . We show: Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m $(α) \leq r_{0} (G) \leq γ \leq 2^{α}$ , or α > ω and $α^{ω} \leq r_{0} (G) \leq 2^{α}$ , then G admits a pseudocompact group topology of weight α. Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies $r_{0} (G) \geq m (α)$ . Theorem 5.2(b). If G is divisible Abelian with $2^{r_{0} (G)} \leq γ$ , then G admits at most $2^{γ}$ -many...

In a left-topological semigroup with dense center the closure of any left ideal is an ideal.

W. Ruppert (1987)

Semigroup forum

In memory of Wiesław Szlenk

(1998)

Fundamenta Mathematicae

In quest of weaker connected topologies

Mihail G. Tkachenko, Vladimir Vladimirovich Tkachuk, Vladimir Vladimirovich Uspenskij, Richard Gordon Wilson (1996)

Commentationes Mathematicae Universitatis Carolinae

We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by “nice” continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather...

In search for Lindelöf $C_{p}$ ’s

Raushan Z. Buzyakova (2004)

Commentationes Mathematicae Universitatis Carolinae

It is shown that if $X$ is a first-countable countably compact subspace of ordinals then $C_{p} (X)$ is Lindelöf. This result is used to construct an example of a countably compact space $X$ such that the extent of $C_{p} (X)$ is less than the Lindelöf number of $C_{p} (X)$ . This example answers negatively Reznichenko’s question whether Baturov’s theorem holds for countably compact spaces.