On Cardinality Bounds Involving the Weak Lindelöf Degree and H-Cl

University of Dayton eCommons Summer Conference on Topology and Its Applications Department of Mathematics 6-2017 On Cardinality Bounds Involving the Weak Lindelöf Degree and H-Closed Spaces Nathan Carlson California Lutheran University, ncarlson@callutheran.edu Angelo Bella Jack Porter Follow this and additional works at: http://ecommons.udayton.edu/topology_conf Part of the Geometry and Topology Commons, and the Special Functions Commons eCommons Citation Carlson, Nathan; Bella, Angelo; and Porter, Jack, "On Cardinality Bounds Involving the Weak Lindelöf Degree and H-Closed Spaces" (2017) Summer Conference on Topology and Its Applications 39 http://ecommons.udayton.edu/topology_conf/39 This Topology + Foundations is brought to you for free and open access by the Department of Mathematics at eCommons It has been accepted for inclusion in Summer Conference on Topology and Its Applications by an authorized administrator of eCommons For more information, please contact frice1@udayton.edu, mschlangen1@udayton.edu Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces On Cardinality Bounds involving the weak Lindelöf degree and H-closed spaces Nathan Carlson California Lutheran University Co-authors: Angelo Bella, Jack Porter 32nd Summer Conference on Topology and its Applications Dayton, June 30, 2017 Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Overview All spaces are Hausdorff Bella and C construct a new closing-off argument and prove the following Theorem Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Overview All spaces are Hausdorff Bella and C construct a new closing-off argument and prove the following Theorem (a) If X is locally compact then |X | ≤ 2wL(X )ψ(X ) Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Overview All spaces are Hausdorff Bella and C construct a new closing-off argument and prove the following Theorem (a) If X is locally compact then |X | ≤ 2wL(X )ψ(X ) (b) If X is regular with a π-base of open sets with compact closures then |X | ≤ 2wL(X )t(X )ψ(X ) Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Overview All spaces are Hausdorff Bella and C construct a new closing-off argument and prove the following Theorem (a) If X is locally compact then |X | ≤ 2wL(X )ψ(X ) (b) If X is regular with a π-base of open sets with compact closures then |X | ≤ 2wL(X )t(X )ψ(X ) (c) If X is locally compact and power homogeneous then |X | ≤ 2wL(X )t(X ) Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Overview All spaces are Hausdorff Bella and C construct a new closing-off argument and prove the following Theorem (a) If X is locally compact then |X | ≤ 2wL(X )ψ(X ) (b) If X is regular with a π-base of open sets with compact closures then |X | ≤ 2wL(X )t(X )ψ(X ) (c) If X is locally compact and power homogeneous then |X | ≤ 2wL(X )t(X ) (d) If X is extremally disconnected then |X | ≤ 2wL(X )πχ (X )ψ(X ) Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Porter and C define a cardinal invariant L(X ) with the properties L(X ) ≤ L(X ), and This gives a common proof of the following for Hausdorff spaces X : Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Porter and C define a cardinal invariant L(X ) with the properties L(X ) ≤ L(X ), and L(X ) is countable if X is H-closed This gives a common proof of the following for Hausdorff spaces X : Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Porter and C define a cardinal invariant L(X ) with the properties L(X ) ≤ L(X ), and L(X ) is countable if X is H-closed Theorem (Porter, C.) If X is Hausdorff then |X | ≤ 2L(X )χ(X ) This gives a common proof of the following for Hausdorff spaces X : Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Proposition Let X be a space, and A, B ⊆ X (a) A ⊆ c(A) (b) if A ⊆ B then c(A) ⊆ c(B) (c) clA ⊆ c(A) ⊆ clθ (A) (d) if U is open, then clU = c(U) ⊆ c(U) Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Proposition Let X be a space, and A, B ⊆ X (a) A ⊆ c(A) (b) if A ⊆ B then c(A) ⊆ c(B) (c) clA ⊆ c(A) ⊆ clθ (A) (d) if U is open, then clU = c(U) ⊆ c(U) (e) if X is regular then clA = c(A) = clθ (A) Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Proposition Let X be a space, and A, B ⊆ X (a) A ⊆ c(A) (b) if A ⊆ B then c(A) ⊆ c(B) (c) clA ⊆ c(A) ⊆ clθ (A) (d) if U is open, then clU = c(U) ⊆ c(U) (e) if X is regular then clA = c(A) = clθ (A) (f) If A is c-closed then A is closed Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Proposition Let X be a space, and A, B ⊆ X (a) A ⊆ c(A) (b) if A ⊆ B then c(A) ⊆ c(B) (c) clA ⊆ c(A) ⊆ clθ (A) (d) if U is open, then clU = c(U) ⊆ c(U) (e) if X is regular then clA = c(A) = clθ (A) (f) If A is c-closed then A is closed (g) c(A) is a closed subset of X Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Proposition Let X be a space, and A, B ⊆ X (a) A ⊆ c(A) (b) if A ⊆ B then c(A) ⊆ c(B) (c) clA ⊆ c(A) ⊆ clθ (A) (d) if U is open, then clU = c(U) ⊆ c(U) (e) if X is regular then clA = c(A) = clθ (A) (f) If A is c-closed then A is closed (g) c(A) is a closed subset of X (h) If X is H-closed then c(A) is an H-set Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Proposition If X is a space and C is a c-closed subset of X , then L(C, X ) ≤ L(X ) I.e., the invariant L(X ) is hereditary on c-closed subsets of X Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Proposition For any Hausdorff space X and for all x = y ∈ X there exist open sets U and V such that x ∈ U, y ∈ V , and U ∩ V = ∅ The above is formally stronger than the usual definition of Hausdorff Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Proposition If X is Hausdorff and A ⊆ X , then |c(A)| ≤ |A|χ(X ) As clA ⊆ c(A), this improves the well-known result |clA| ≤ |A|χ(X ) Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Using the previous results, the operator c, and a standard closing off-argument, we obtain: Main Theorem (C., Porter, 2016) If X is Hausdorff then |X | ≤ 2L(X )χ(X ) As L(X ) = ℵ0 for an H-closed space X , it follows that: Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Using the previous results, the operator c, and a standard closing off-argument, we obtain: Main Theorem (C., Porter, 2016) If X is Hausdorff then |X | ≤ 2L(X )χ(X ) As L(X ) = ℵ0 for an H-closed space X , it follows that: Corollary (Dow, Porter 1982) If X is H-closed then |X | ≤ 2χ(X ) (in fact, |X | ≤ 2ψc (X ) ) Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces We can now identify a property P of a Hausdorff space X that generalizes both the H-closed and Lindelöf properties such that |X | ≤ 2χ(X ) for spaces with property P: P = for every open cover V of X there is W ∈ [V]≤ω such that X = W W ∈W Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Question Is L(X ) independent of the choice of open ultrafilter assignment? Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces Thank you! Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces A Bella, N A Carlson, On cardinality bounds involving the weak Lindelöf degree, to appear in Quaestiones Mathematicae Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces A Bella, N A Carlson, On cardinality bounds involving the weak Lindelöf degree, to appear in Quaestiones Mathematicae N A Carlson, J.R Porter On the cardinality of Hausdorff spaces and H-closed spaces, Topology Appl (2017), DOI Nathan Carlson wL(X ) and H-closed spaces ... ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality bound for Hausdorff spaces On Cardinality Bounds involving the weak Lindelöf degree. .. Bella and C construct a new closing-off argument and prove the following Theorem Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the. .. compact then |X | ≤ 2wL(X )ψ(X ) Nathan Carlson wL(X ) and H-closed spaces Overview wL(X ) Cardinality bounds involving wL(X ) H-closed spaces U, the operator c, and the invariant L(X ) A cardinality