Let $F(X)$ be the supremum of cardinalities of free sequences in $X$. We prove that the radial ch... more Let $F(X)$ be the supremum of cardinalities of free sequences in $X$. We prove that the radial character of every Lindelof Hausdorff almost radial space $X$ and the set-tightness of every Lindelof Hausdorff space are always bounded above by $F(X)$. Solving a question of Bella, we exhibit a Hausdorff radial space $X$ whose radial character is strictly larger than $F(X)$. We then improve a result of Dow, Juhasz, Soukup, Szentmiklossy and Weiss by proving that if $X$ is a Lindelof Hausdorff space, and $X_\delta$ denotes the $G_\delta$ topology on $X$ then $t(X_\delta) \leq 2^{t(X)}$. Finally, we exploit this to prove that if $X$ is a Lindelof Hausdorff pseudoradial space then $F(X_\delta) \leq 2^{F(X)}$, which partially answer a question of Bella and ourselves.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
We establish several bounds on the cardinality of a topological space involving the Hausdorff pse... more We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter $$H\psi (X)$$ H ψ ( X ) . This invariant has the property $$\psi _c(X)\le H\psi (X)\le \chi (X)$$ ψ c ( X ) ≤ H ψ ( X ) ≤ χ ( X ) for a Hausdorff space X. We show the cardinality of a Hausdorff space X is bounded by $$2^{pwL_c(X)H\psi (X)}$$ 2 p w L c ( X ) H ψ ( X ) , where $$pwL_c(X)\le L(X)$$ p w L c ( X ) ≤ L ( X ) and $$pwL_c(X)\le c(X)$$ p w L c ( X ) ≤ c ( X ) . This generalizes results of Bella and Spadaro, as well as Hodel. We show additionally that if X is a Hausdorff linearly Lindelöf space such that $$H\psi (X)=\omega $$ H ψ ( X ) = ω , then $$|X|\le 2^\omega $$ | X | ≤ 2 ω , under the assumption that either $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ 2 < c = c or $${\mathfrak {c}}<\aleph _\omega $$ c < ℵ ω . The following game-theoretic result is shown: if X is a regular space such that player two has a winning strategy in the game $$G^{\kappa }_1({\mat...
We construct a consistent example of a topological space $$Y= {X \cup \{\infty\}} $$ Y = X ∪ { ∞ ... more We construct a consistent example of a topological space $$Y= {X \cup \{\infty\}} $$ Y = X ∪ { ∞ } such that: (1) $$Y$$ Y is regular. (2) Every $$G_\delta$$ G δ subset of $$Y$$ Y is open. (3) The point $$\infty$$ ∞ is not isolated, but it is not in the closure of any discrete subset of $$X$$ X .
We solve a long standing question due to Arhangel'skii by constructing a compact space which ... more We solve a long standing question due to Arhangel'skii by constructing a compact space which has a G_δ cover with no continuum-sized (G_δ)-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every G_δ cover has a c-sized subcollection with a G_δ-dense union and that in a Lindelöf space with a base of multiplicity continuum, every G_δ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Vega's celebrated theorem on the cardinality of homogeneous compacta of countable tightness.
We prove an upper bound on the tightness of the G_δ-modification of any regular space. Namely, if... more We prove an upper bound on the tightness of the G_δ-modification of any regular space. Namely, if X_δ denotes the G_δ-modification of the space X, we prove that t(X_δ) ≤ 2^F(X) where F(X) denotes the supremum of cardinalities of free sequences in X. This is a substantial improvement of the bound t(X_δ) ≤ 2^t(X), proved for every Lindelöf regular space X by Dow, Juhász, Soukup, Szentmiklóssy and Weiss. As a byproduct we obtain the natural bound F(X_δ) ≤ 2^F(X) for the length of the free sequences in the G_δ topology of a Lindelöf regular space X.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2021
We show that if X is a first-countable Urysohn space where player II has a winning strategy in th... more We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game $$G^{\omega _1}_{fin}({\mathcal {O}}, {\mathcal {O}}_D)$$ G fin ω 1 ( O , O D ) , providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelöf game of length $$\omega _1$$ ω 1 .
We solve two questions regarding spaces with a (Gδ)-diagonal of rank 2. One is a question of Basi... more We solve two questions regarding spaces with a (Gδ)-diagonal of rank 2. One is a question of Basile, Bella and Ridderbos about weakly Lindelöf spaces with a Gδ-diagonal of rank 2 and the other is a question of Arhangel’skii and Bella asking whether every space with a diagonal of rank 2 and cellularity continuum has cardinality at most continuum.
We present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies v... more We present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\leqslant 2^{L(X)\unicode[STIX]{x1D712}(X)}$ (Arhangel’skiĭ) and $|X|\leqslant 2^{c(X)\unicode[STIX]{x1D712}(X)}$ (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s solution to Alexandroff’s problem: A survey, Topology Appl. 153(2006), 2199–2217).In contrast to previous attempts, we do not need any separation axiom beyond $T_{2}$.
Let $F(X)$ be the supremum of cardinalities of free sequences in $X$. We prove that the radial ch... more Let $F(X)$ be the supremum of cardinalities of free sequences in $X$. We prove that the radial character of every Lindelof Hausdorff almost radial space $X$ and the set-tightness of every Lindelof Hausdorff space are always bounded above by $F(X)$. Solving a question of Bella, we exhibit a Hausdorff radial space $X$ whose radial character is strictly larger than $F(X)$. We then improve a result of Dow, Juhasz, Soukup, Szentmiklossy and Weiss by proving that if $X$ is a Lindelof Hausdorff space, and $X_\delta$ denotes the $G_\delta$ topology on $X$ then $t(X_\delta) \leq 2^{t(X)}$. Finally, we exploit this to prove that if $X$ is a Lindelof Hausdorff pseudoradial space then $F(X_\delta) \leq 2^{F(X)}$, which partially answer a question of Bella and ourselves.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
We establish several bounds on the cardinality of a topological space involving the Hausdorff pse... more We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter $$H\psi (X)$$ H ψ ( X ) . This invariant has the property $$\psi _c(X)\le H\psi (X)\le \chi (X)$$ ψ c ( X ) ≤ H ψ ( X ) ≤ χ ( X ) for a Hausdorff space X. We show the cardinality of a Hausdorff space X is bounded by $$2^{pwL_c(X)H\psi (X)}$$ 2 p w L c ( X ) H ψ ( X ) , where $$pwL_c(X)\le L(X)$$ p w L c ( X ) ≤ L ( X ) and $$pwL_c(X)\le c(X)$$ p w L c ( X ) ≤ c ( X ) . This generalizes results of Bella and Spadaro, as well as Hodel. We show additionally that if X is a Hausdorff linearly Lindelöf space such that $$H\psi (X)=\omega $$ H ψ ( X ) = ω , then $$|X|\le 2^\omega $$ | X | ≤ 2 ω , under the assumption that either $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ 2 < c = c or $${\mathfrak {c}}<\aleph _\omega $$ c < ℵ ω . The following game-theoretic result is shown: if X is a regular space such that player two has a winning strategy in the game $$G^{\kappa }_1({\mat...
We construct a consistent example of a topological space $$Y= {X \cup \{\infty\}} $$ Y = X ∪ { ∞ ... more We construct a consistent example of a topological space $$Y= {X \cup \{\infty\}} $$ Y = X ∪ { ∞ } such that: (1) $$Y$$ Y is regular. (2) Every $$G_\delta$$ G δ subset of $$Y$$ Y is open. (3) The point $$\infty$$ ∞ is not isolated, but it is not in the closure of any discrete subset of $$X$$ X .
We solve a long standing question due to Arhangel'skii by constructing a compact space which ... more We solve a long standing question due to Arhangel'skii by constructing a compact space which has a G_δ cover with no continuum-sized (G_δ)-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every G_δ cover has a c-sized subcollection with a G_δ-dense union and that in a Lindelöf space with a base of multiplicity continuum, every G_δ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Vega's celebrated theorem on the cardinality of homogeneous compacta of countable tightness.
We prove an upper bound on the tightness of the G_δ-modification of any regular space. Namely, if... more We prove an upper bound on the tightness of the G_δ-modification of any regular space. Namely, if X_δ denotes the G_δ-modification of the space X, we prove that t(X_δ) ≤ 2^F(X) where F(X) denotes the supremum of cardinalities of free sequences in X. This is a substantial improvement of the bound t(X_δ) ≤ 2^t(X), proved for every Lindelöf regular space X by Dow, Juhász, Soukup, Szentmiklóssy and Weiss. As a byproduct we obtain the natural bound F(X_δ) ≤ 2^F(X) for the length of the free sequences in the G_δ topology of a Lindelöf regular space X.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2021
We show that if X is a first-countable Urysohn space where player II has a winning strategy in th... more We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game $$G^{\omega _1}_{fin}({\mathcal {O}}, {\mathcal {O}}_D)$$ G fin ω 1 ( O , O D ) , providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelöf game of length $$\omega _1$$ ω 1 .
We solve two questions regarding spaces with a (Gδ)-diagonal of rank 2. One is a question of Basi... more We solve two questions regarding spaces with a (Gδ)-diagonal of rank 2. One is a question of Basile, Bella and Ridderbos about weakly Lindelöf spaces with a Gδ-diagonal of rank 2 and the other is a question of Arhangel’skii and Bella asking whether every space with a diagonal of rank 2 and cellularity continuum has cardinality at most continuum.
We present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies v... more We present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\leqslant 2^{L(X)\unicode[STIX]{x1D712}(X)}$ (Arhangel’skiĭ) and $|X|\leqslant 2^{c(X)\unicode[STIX]{x1D712}(X)}$ (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s solution to Alexandroff’s problem: A survey, Topology Appl. 153(2006), 2199–2217).In contrast to previous attempts, we do not need any separation axiom beyond $T_{2}$.
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