Regular $${G_\delta}$$ G δ -diagonals and some upper bounds for cardinality of topological spaces

Abstract

We prove that, under CH, any space with a regular G δ -diagonal and caliber ω1 is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space X, we establish the inequality |X| ≤ wL(X) s∆ 2 (X)·dot(X) which represents a generalization of a theorem of Basile, Bella, and Ridderbos. We also show that if X is a Hausdorff space, then |X| ≤ (πχ(X) · d(X)) ot(X)·ψc(X) ; this result im-pliesŠapirovskiȋ's inequality |X| ≤ πχ(X) c(X)·ψ(X) which only holds for regular spaces. It is also proved that |X| ≤ πχ(X) ot(X)·ψc(X)·aLc(X) for any Hausdorff space X; this gives one more generalization of the famous Arhangel ′ skii's inequality |X| ≤ 2 χ(X)·L(X) .

New bounds on the cardinality of Hausdorff spaces and regular spaces

“…In (1), ot(X) ≤ wt(X) was originally shown in [4]. (2) was mentioned in [13], (3) was mentioned in [24], (4), (5), and ( 6) are new in this paper, and (7) was mentioned in [16]. For (8), |RO(X)| ≤ πw(X) c(X) is due to Efimov [11], and |RO(X)| ≤ 2 d(X) is due to de Groot [14].…”

“…The cardinal function ot(X) was defined by Tkachenko in [24] and used by Gotchev, Tkachenko, and Tkachuk in [13] as well as by Bella, the author, and Gotchev in [4]. The related function dot(X) was defined in [13].…”

“…The cardinal function ot(X) was defined by Tkachenko in [24] and used by Gotchev, Tkachenko, and Tkachuk in [13] as well as by Bella, the author, and Gotchev in [4]. The related function dot(X) was defined in [13]. This function should be considered "small" in a sense, as dot(X) ≤ min{ot(X), πχ(X)} for any space X (Proposition 2.9(1)(2)).…”

“…ψ(X) cannot be replaced with dψ c (X)wψ c (X) in the Arhangel ′ skiȋ-Šapirovskiȋ bound 2 L(X)t(X)ψ(X) for Hausdorff spaces. Moreover, it was shown in [13] that the cardinality of a Hausdorff space is always bounded by πχ(X) aLc(X)ot(X)ψc (X) , where aL c (X), the almost Lindelöf degree with respect to closed sets, satisfies aL c (X) ≤ L(X). Our compact space X is also a counterexample to replacing ψ c (X) with dψ c (X)wψ c (X) in this bound.…”

“…In [13] it was shown by Gotchev, Tkachenko, and Tkachuk that if X is Hausdorff then |X| ≤ πw(X) ot(X)ψc (X) . We show in the next result that ψ c (X) can be replaced with wψ c (X).…”