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

Increased Rotatory Laxity after Anterolateral Ligament Lesion in Anterior Cruciate Ligament- (ACL-) Deficient Knees: A Cadaveric Study with Noninvasive Inertial Sensors

The anterolateral ligament (ALL) has been suggested as an important secondary knee restrain on the dynamic laxity in anterior cruciate ligament- (ACL-) deficient knees. Nevertheless, its kinematical contribution to the pivot-shift (PS) phenomenon has not been clearly and objectively defined, and noninvasive sensor technology could give a crucial contribution in this direction. The aim of the present study was to quantify in vitro the PS phenomenon in order to investigate the differences between an ACL-deficient knee and an ACL+ALL-deficient knee. Ten fresh-frozen paired human cadaveric knees (n=20) were included in this controlled laboratory study. Intact, ACL-deficient, and ACL+ALL-deficient knees were subjected to a manual PS test quantified by a noninvasive triaxial accelerometer (KiRA, OrthoKey). Kinematic data (i.e., posterior acceleration of the tibial lateral compartment) were recorded and compared among the three statuses. Pairwise Student's t-test was used to compare the single groups (p<0.05). Intact knees, ACL-deficient knees, and ACL+ALL-deficient knees showed an acceleration of 5.3±2.1 m/s2, 6.3±2.3 m/s2, and 7.8±2.1 m/s2, respectively. Combined sectioning of ACL and ALL resulted in a statistically significant acceleration increase compared to both the intact state (p<0.01) and the ACL-deficient state (p<0.01). The acceleration increase determined by isolated ACL resection compared to the intact state was not statistically significant (p>0.05). The ALL sectioning increased the rotatory laxity during the PS after ACL sectioning as measured through a user-friendly, noninvasive triaxial accelerometer

Segond's fracture: a biomechanical cadaveric study using navigation

Background Segond’s fracture is a well-recognised radiological sign of an anterior cruciate ligament (ACL) tear. While previous studies evaluated the role of the anterolateral ligament (ALL) and complex injuries on rotational stability of the knee, there are no studies on the biomechanical effect of Segond’s fracture in an ACL deficient knee. The aim of this study was to evaluate the effect of a Segond’s fracture on knee rotation stability as evaluated by a navigation system in an ACL deficient knee. Materials and methods Three different conditions were tested on seven knee specimens: intact knee, ACL deficient knee and ACL deficient knee with Segond’s fracture. Static and dynamic measurements of anterior tibial translation (ATT) and axial tibial rotation (ATR) were recorded by the navigation system (2.2 OrthoPilot ACL navigation system B. Braun Aesculap, Tuttlingen, Germany). Results Static measurements at 30 showed that the mean ATT at 30 of knee flexion was 5.1 ± 2.7 mm in the ACL intact condition, 14.3 ± 3.1 mm after ACL cut (P = 0.005), and 15.2 ± 3.6 mm after Segond’s fracture (P = 0.08). The mean ATR at 30 of knee flexion was 20.7 ± 4.8 in the ACL intact condition, 26.9 ± 4.1 in the ACL deficient knee (P[0.05) and 30.9 ± 3.8 after Segond’s fracture (P = 0.005). Dynamic measurements during the pivot-shift showed that the mean ATT was 7.2 ± 2.7 mm in the intact knee, 9.1 ± 3.3 mm in the ACL deficient knee(P = 0.04) and 9.7 ± 4.3 mm in the ACL deficient knee with Segond’s fracture (P = 0.07). The mean ATR was 9.6 ± 1.8 in the intact knee, 12.3 ± 2.3 in the ACL deficient knee (P[0.05) and 19.1 ± 3.1 in the ACL deficient knee with Segond’s lesion (P = 0.016). Conclusion An isolated lesion of the ACL only affects ATT during static and dynamic measurements, while the addition of Segond’s fracture has a significant effect on ATR in both static and dynamic execution of the pivot-shift test, as evaluated with the aid of navigation

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

The character of topological groups, via bounded systems, Pontryagin--van Kampen duality and pcf theory

The Birkhoff--Kakutani Theorem asserts that a topological group is metrizable if and only if it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups. We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are the closed subgroups of Pontryagin--van Kampen duals of \emph{metrizable} abelian groups, or equivalently, complete abelian groups whose dual is metrizable. By investigating these connections, we show that also in these cases, the character can be estimated, and that it is determined by the weights of the \emph{compact} subsets of the group, or of quotients of the group by compact subgroups. It follows, for example, that the density and the local density of an abelian metrizable group determine the character of its dual group. Our main result applies to the more general case of closed subgroups of Pontryagin--van Kampen duals of abelian \v{C}ech-complete groups. In the special case of free abelian topological groups, our results extend a number of results of Nickolas and Tkachenko, which were proved using combinatorial methods. In order to obtain concrete estimations, we establish a natural bridge between the studied concepts and pcf theory, that allows the direct application of several major results from that theory. We include an introduction to these results and their use.Comment: Minor corrections. Final version before journal editin

Preservation and reflection of properties acc and hacc

summary:The aim of the paper is to study the preservation and the reflection of acc and hacc spaces under various kinds of mappings. In particular, we show that acc and hacc are not preserved by perfect mappings and that acc is not reflected by closed (nor perfect) mappings while hacc is reflected by perfect mappings

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

Ricostruzione con STG a Singolo Fascio – Tecnica Transtibiale, Transportale, Over The Top

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