Compactness-like Covering Properties Petra Staynova Durham University November 7, 2013 Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 1 / 31
Something to think about... [Look at the board] Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 2 / 31
Introduction One of the main generalisations of compactness is the notion of an H-closed space: Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 3 / 31
Introduction One of the main generalisations of compactness is the notion of an H-closed space: Definition (H-closed, general) A topological space X is said to be H-closed iff it is closed in every Hausdorff space containing it as a subspace. Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 3 / 31
Introduction One of the main generalisations of compactness is the notion of an H-closed space: Definition (H-closed, general) A topological space X is said to be H-closed iff it is closed in every Hausdorff space containing it as a subspace. A more useful definition is: Definition (H-closed) A topological space X is said to be H-closed iff every open cover has a finite subfamily with dense union. Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 3 / 31
Countable generalisations Another generalisation of compactness is the well-known Lindel¨ of property: Definition (Lindel¨ of) A topological space X is called Lindel¨ of iff every open cover has a countable subcover. Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31
Countable generalisations Another generalisation of compactness is the well-known Lindel¨ of property: Definition (Lindel¨ of) A topological space X is called Lindel¨ of iff every open cover has a countable subcover. Example (Lindel¨ of spaces) The following spaces are Lindel¨ of: Any countable topological space; Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31
Countable generalisations Another generalisation of compactness is the well-known Lindel¨ of property: Definition (Lindel¨ of) A topological space X is called Lindel¨ of iff every open cover has a countable subcover. Example (Lindel¨ of spaces) The following spaces are Lindel¨ of: Any countable topological space; Any space with the co-countable topology; Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31
Countable generalisations Another generalisation of compactness is the well-known Lindel¨ of property: Definition (Lindel¨ of) A topological space X is called Lindel¨ of iff every open cover has a countable subcover. Example (Lindel¨ of spaces) The following spaces are Lindel¨ of: Any countable topological space; Any space with the co-countable topology; A countable union of compact spaces; Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31
Countable generalisations Another generalisation of compactness is the well-known Lindel¨ of property: Definition (Lindel¨ of) A topological space X is called Lindel¨ of iff every open cover has a countable subcover. Example (Lindel¨ of spaces) The following spaces are Lindel¨ of: Any countable topological space; Any space with the co-countable topology; A countable union of compact spaces; R , with the Euclidean topology. Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31
Countable generalisations Another generalisation of compactness is the well-known Lindel¨ of property: Definition (Lindel¨ of) A topological space X is called Lindel¨ of iff every open cover has a countable subcover. Example (Lindel¨ of spaces) The following spaces are Lindel¨ of: Any countable topological space; Any space with the co-countable topology; A countable union of compact spaces; R , with the Euclidean topology. The Sorgenfrey line S ( R with the topology generated by the base B = { [ a , b ) : a < b } ). Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31
Countable generalisations, cont’d In 1959, Zdenek Frolik introduced a notion that combines the Lindel¨ of and H-closed properties: Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 5 / 31
Countable generalisations, cont’d In 1959, Zdenek Frolik introduced a notion that combines the Lindel¨ of and H-closed properties: Definition (weakly Lindel¨ of , [Fro59]) A topological space X is weakly Lindel¨ of if for every open cover U of X there is a countable subfamily U ′ ⊆ U such that X = � U ′ . Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 5 / 31
Countable generalisations, cont’d In 1959, Zdenek Frolik introduced a notion that combines the Lindel¨ of and H-closed properties: Definition (weakly Lindel¨ of , [Fro59]) A topological space X is weakly Lindel¨ of if for every open cover U of X there is a countable subfamily U ′ ⊆ U such that X = � U ′ . Later on, while studying cardinal invariants, Dissanayeke and Willard introduced another generalisation of the Lindel¨ of property, which is stronger than the weakly Lindel¨ of one: Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 5 / 31
Countable generalisations, cont’d In 1959, Zdenek Frolik introduced a notion that combines the Lindel¨ of and H-closed properties: Definition (weakly Lindel¨ of , [Fro59]) A topological space X is weakly Lindel¨ of if for every open cover U of X there is a countable subfamily U ′ ⊆ U such that X = � U ′ . Later on, while studying cardinal invariants, Dissanayeke and Willard introduced another generalisation of the Lindel¨ of property, which is stronger than the weakly Lindel¨ of one: Definition (almost Lindel¨ of , [WD84]) of iff for every open cover U of X A topological space X is almost Lindel¨ there is a countable subfamily U ′ ⊆ U such that X = � { U : U ∈ U ′ } . Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 5 / 31
Generalisations of compactness: a diagram compact keep finiteness H-closed keep requirement for cover almost weakly Lindelof Lindelof Lindelof Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 6 / 31
Separation Axioms T0 T={[a, infty) for a in R} - topology on [0, infty) T1 Hausdorff Urysohn More complex Functionally Hausdorff +locallly compact Regular Functionally Regular = Tychonoff Normal Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 7 / 31
Properties of the generalisations Proposition Every regular Hausdorff H-closed space is compact. Every regular Lindel¨ of space is normal. Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 8 / 31
Properties of the generalisations Proposition Every regular Hausdorff H-closed space is compact. Every regular Lindel¨ of space is normal. Proposition A regular almost Lindel¨ of space is Lindel¨ of. Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 8 / 31
Properties of the generalisations Proposition Every regular Hausdorff H-closed space is compact. Every regular Lindel¨ of space is normal. Proposition A regular almost Lindel¨ of space is Lindel¨ of. Proposition A normal weakly Lindel¨ of space is almost Lindel¨ of. Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 8 / 31
Properties of the generalisations Proposition The continuous image of an almost Lindel¨ of (weakly Lindel¨ of) space is almost Lindel¨ of (weakly Lindel¨ of). Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 9 / 31
Properties of the generalisations Proposition The continuous image of an almost Lindel¨ of (weakly Lindel¨ of) space is almost Lindel¨ of (weakly Lindel¨ of). Proposition A clopen subset of an almost Lindel¨ of (weakly Lindel¨ of) space is almost Lindel¨ of (weakly Lindel¨ of). Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 9 / 31
Properties of the generalisations Proposition The continuous image of an almost Lindel¨ of (weakly Lindel¨ of) space is almost Lindel¨ of (weakly Lindel¨ of). Proposition A clopen subset of an almost Lindel¨ of (weakly Lindel¨ of) space is almost Lindel¨ of (weakly Lindel¨ of). Proposition of) and Y is compact, then X × Y is If X is almost Lindel¨ of (weakly Lindel¨ almost Lindel¨ of (weakly Lindel¨ of). Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 9 / 31
Inheritance Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 10 / 31
A modification Definition (quasi-Lindel¨ of , [Arh79]) We call a space X quasi-Lindel¨ of if for every closed subset Y of X and every collection U of open in X sets such that Y ⊆ � U , there is a countable subfamily U ′ ⊆ U such that Y ⊂ � U ′ . Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 11 / 31
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