More on the cardinality of a topological space

In this paper we continue to investigate the impact that various separation axioms and covering properties have onto the cardinality of topological spaces. Many authors have been working in that field. To mention a few, let us refer to results by Arhangel’skii, Alas, Hajnal-Juhász, Bell-Gisburg-Woods, Dissanayake-Willard, Schröder and to the excellent survey by Hodel “Arhangel’skii’s Solution to Alexandroff’s problem: A survey”. Here we provide improvements and analogues of some of the results obtained by the above authors in the settings of more general separation axioms and cardinal invariants related to them. We also provide partial answer to Arhangel’skii’s question concerning whether the continuum is an upper bound for the cardinality of a Hausdorff Lindelöf space having countable pseudo-character (i.e., points are Gδ). Shelah in 1978 was the first to give a consistent negative answer to Arhangel’skii’s question; in 1993 Gorelic established an improved result; and further results were obtained by Tall in 1995.  The question of whether or not there is a consistent bound on the cardinality of Hausdorff Lindelöf spaces with countable pseudo-character is still open. In this paper we introduce the Hausdorff point separating weight Hpw(X), and prove that (1) |X| ≤ Hpsw(X)aLc(X)ψ(X), for Hausdorff spaces and (2) |X| ≤ Hpsw(X)ωLc(X)ψ(X), where X is a Hausdorff space with a π-base consisting of compact sets with non-empty interior. In 1993 Schröder proved an analogue of Hajnal and Juhasz inequality |X| ≤ 2c(X)χ(X) for Hausdorff spaces, for Urysohn spaces by considering weaker invariant - Urysohn cellularity Uc(X) instead of cellularity c(X). We introduce the n-Urysohn cellularity n-Uc(X) (where n≥2) and prove that the previous inequality is true in the class of n-Urysohn spaces replacing Uc(X) by the weaker n-Uc(X). We also show that |X| ≤ 2Uc(X)πχ(X) if X is a power homogeneous Urysohn space

Anatomic and histological study of the anterolateral aspect of the knee: a SANTI Group investigation

Background: The structure and function of the anterolateral aspect of the knee have been significantly debated, with renewed interest in this topic since the description of the anterolateral ligament (ALL). Purpose: To define and describe the distinct structures of the lateral knee and to correlate the macroscopic and histologic anatomic features. Study Design: Descriptive laboratory study. Methods: Twelve fresh-frozen human cadavers were used for anatomic analysis. In the left knee, a layer-by-layer dissection and macroscopic analysis were performed. In the right knee, an en bloc specimen was obtained encompassing an area from the Gerdy tubercle to the posterior fibular head and extending proximally from the anterior aspect to the posterior aspect of the lateral femoral epicondyle. The en bloc resection was then frozen, sliced at the level of the joint line, and reviewed by a musculoskeletal pathologist. Results: Macroscopically, the lateral knee has 4 main layers overlying the capsule of the knee: the aponeurotic layer, the superficial layer including the iliotibial band (ITB), the deep fascial layer, and the ALL. Histologically, 8 of 12 specimens demonstrated 4 consistent, distinct structures: the ITB, the ALL, the lateral collateral ligament, and the meniscus. Conclusion: The lateral knee has a complex orientation of layers and fibers. The ALL is a distinct structure from the ITB and is synonymous to the previously described capsulo-osseous layer of the ITB. Clinical Relevance: Increasingly, lateral extra-articular procedures are performed at the time of anterior cruciate ligament reconstruction. Understanding the anatomic features of the anterolateral aspect of the knee is necessary to understand the biomechanics and function of the structures present and allows surgeons to attempt to replicate those anatomic characteristics when performing extra-articular reconstruction

Centered-Lindelöfness versus star-Lindelöfness

summary:We discuss various generalizations of the class of Lindelöf spaces and study the difference between two of these generalizations, the classes of star-Lindelöf and centered-Lindelöf spaces

On a weaker form of countable compactness

A star covering property which is equivalent to countable compactness for regular spaces and weaker than countable compactness for Hausdorff spaces is introduced and considered. Various kinds of irregularity of topological spaces are discussedQuaestiones Mathematicae 30(2007), 407–41