T1-space Research Articles

Existence of well-filterifications of T0 topological spaces

We prove that for every T0 space X, there is a well-filtered space W(X) and a continuous mapping ηX:X⟶W(X), such that for any well-filtered space Y and any continuous mapping f:X⟶Y there is a unique continuous mapping fˆ:W(X)⟶Y such that f=fˆ∘ηX. Such a space W(X) will be called the well-filterification of X. This result gives a positive answer to one of the major open problems on well-filtered spaces. As a corollary, we obtain that the product of well-filtered spaces is well-filtered.

The congruence biframe as a quasi-uniform bicompletion

Künzi and Ferrario have shown that a T0 space is sober if and only if it is bicomplete in the well-monotone quasi-uniformity. We prove a pointfree version of this result: a strictly zero-dimensional biframe is a congruence biframe if and only if it is bicomplete in the same quasi-uniformity. As a corollary we obtain a new proof of a result of Plewe that a congruence frame is ultraparacompact. The main result makes use of a new construction of the bicompletion of a quasi-uniform biframe as a quotient of the Samuel compactification.

An Examination of P. Davidson’s Misunderstandings about the Limiting Frequency Theory of Probability, Ergodicity and Non Ergodicity: These Approaches Require that either the Number of Observations Approaches Infinity or Time Approaches Infinity in Order for the Limit to Exist

Paul Davidson’s technical understanding of the mathematical details of the Limiting Frequency theory of probability and Kolmogorov’s measure theoretic extension from additivity to countable additivity, which allows for an extension of the concept of the Law of Large Numbers to the concept or ergodicity and non ergodicity, where time is substituted for observations, is extremely poor. Davidson has repeatedly made errors in many journal articles published in the Journal of Post Keynesian Economics and books published by Edward Elgar about these concepts since 1979. For instance, one example of a fundamental error made by Davidson since 1982 is his assertion, contradicted by all mathematical statisticians, that “Non stationarity is a sufficient, but not a necessary, condition for nonergodicity”. Davidson has repeated this false claim many times in the literature since 1982 when it first appeared in an article published in the Journal of Post Keynesian Economics. It is very simple to grasp what Davidson’s fundamental error is from just the title of a 1973 paper published in The Annals of Probability by R W Madsen and D L Isaacson, titled “Strongly Ergodic Behavior for Non-Stationary Markov Processes”, to realize that the correct mathematical statement is that non-stationary processes can be ergodic. This means that the correct analysis, known for certain by all mathematical statisticians since 1973, is that “Non stationarity is a necessary, but not a sufficient, condition for nonergodicity”. There is no existing evidence that Paul Davidson had any knowledge of Keynes’s approach to measurement, which rejected the exact and precise approach of Kolmogorov and Tinbergen except as a very special case, in favor of an inexact approach to measurement based on approximation and imprecise, interval valued probability and outcomes. Given that the concepts of ergodicity and non ergodicity require as a necessary condition the acceptance of the measure theoretic approach of Kolmogorov,it is straightforward to conclude that Keynes would reject the concepts of ergodicity and non ergodicity in macroeconomics or in regard to the concept of uncertainty because Part II of the TP establishes non additivity and non linearity as the general case and additivity and linearity as a special case, which is what the Kolmogorov axioms require.

When Intrinsic Randomness Could Come From the Finite/Infinite Transition

Quantum physics is non-causal, and randomness is so-called “intrinsic”. We propose no less than an 18th interpretation of it through non-Archimedean geometry to bring back causality, respect of the Kolmogorov axioms and the existence of hidden variables. For these latter ones, we show that they cannot be in any Hilbert space and hence could not be detected in any traditional experiment. We end through proposing two experiments which would prove the non-Archimedean nature of our universe. The first one consists in a new disruptive type of quantum radar. The second one explains how viscosity naturally occurs in fluid mechanics whereas Boltzmann’s approach only considers elastic shocks at the molecular scale.

Role of contrast-enhanced magnetic resonance high-resolution variable flip angle turbo-spin-echo (T1 SPACE) technique in diagnosis of transverse sinus stenosis

PurposeTransverse sinus stenosis (TSS) is the most sensitive imaging characteristic of idiopathic intracranial hypertension (IIH). This study aimed to assess the diagnostic performance of contrast-enhanced magnetic resonance high-resolution variable flip angle turbo-spin-echo (T1 SPACE) technique in TSS patients and evaluate the diagnostic accuracy of enhanced T1 SPACE, and phase-contrast magnetic resonance venography (PC MRV) with digital subtraction angiography (DSA) as standard imaging. MethodThis prospective study enrolled 62 patients with suspected IIH and PC MRV-confirmed transverse sinus stenosis. All patients underwent lumbar puncture, PC MRV, enhanced T1 SPACE sequences and DSA examination. The accuracy, sensitivity, and specificity of enhanced T1 SPACE in detecting venous sinus stenosis were calculated and compared with those of PC MRV. Intermodality agreement (Kendall’s rank correlation coefficients and weighted kappa statistic) was assessed. ResultsSixty-two patients were enrolled from November 2016 to October 2018. For the measured stenosis, better correlation was observed in enhanced T1 SPACE and DSA (AUC = 0.953) than PC MRV (AUC = 0.871). Intermodality agreement of enhanced T1 SPACE (rk = 0.895 and weighted ĸ = 0.868) was better than PC MRV (rk = 0.753 and weighted ĸ = 0.653) compared with DSA. Thirty-seven intrasinus filling defects were detected by contrast-enhanced T1 SPACE, while only twenty of them were detected on source imaging of PC MRV. ConclusionsThe contrast-enhanced T1 SPACE sequence was more sensitive and specific compared with PC MRV in assessing stenosis and detecting lesions in TSS patients. Accurate determination of the presence and extent of TSS using this technique might be useful in patient selection and guiding the treatment.

More cardinal inequalities in T1/Urysohn spaces

In this paper we provide a partial positive answer to the question 2 in [8]: Do we have |X|≤2L(X)Fa(X)ψ(X), for every T2–space X? Precisely, in Theorem 2.6 we show that |X|≤2L(X)Fa(X)ψθ(X) for Urysohn spaces. Even more, we use cardinal functions recently introduced by Basile, Bonanzinga and Carlson in [4] to generalize some cardinal inequalities; among others, in the Theorem 2.8, we provide a result related with the well known inequality of Hajnal and Juhász: |X|≤2c(X)χ(X); and in the Theorem 2.10, we generalize the inequality |X|≤2wLc(X)χ(X), for any Urysohn space X; due to Ofelia Alas ([1]). Some proofs of the theorems provided in this paper use the elementary submodels technique.

The infinite epistemic regress problem has no unique solution

In this article we analyze the claim that a probabilistic interpretation of the infinite epistemic regress problem leads to a unique solution, the so called “completion” of the regress. This claim is implicitly based on the assumption that the standard Kolmogorov axioms of probability theory are suitable for describing epistemic probability. This assumption, however, has been challenged in the literature, by various authors. One of the alternatives that have been suggested to replace the Kolmogorov axioms in case of an epistemic interpretation of probability, are belief functions, introduced by Shafer in 1976. We show that when one uses belief functions to describe the infinite epistemic regress problem, it is no longer the case that the solution is unique. We also argue that this complies with common sense.

On well-filtered reflections of T0 spaces

Following Ershov's method of constructing the d-completion of T0 spaces, we give a direct construction of the well-filtered reflection of T0 spaces. Also, we obtain an elegant characterization of well-filtered spaces using KF-sets. We then show that a product of a family of T0 spaces is well-filtered iff each factor space is well-filtered. Finally, we obtain that the well-filtered reflection of a product of a finite family of T0 spaces is the product of the well-filtered reflections of all factor spaces. A common theme is that KF-sets, introduced by the first and the fourth authors, function prominently in all of the above proofs.

Signal suppressions of grape syrup and grape syrup/lemon aqueous solutions in magnetic resonance cholangiopancreatography using heavily T2-weighted pulse sequence

Abstract Signal suppression of the gastrointestinal tract fluids in magnetic resonance cholangiopancreatography (MRCP) has been performed using various natural and chemical substances with the different MRCP pulse sequences. This study aimed to investigate the signal suppressions of the grape syrup and the grape syrup/lemon aqueous solutions in MRCP using the heavily T2-weighted sequences. For this purpose, the potassium, iron and manganese contents of grape syrup were measured by atomic absorption spectroscopy. The grape syrup and the grape syrup/lemon solutions with the various grape syrup percentages were imaged using the heavily T2-weighted sequences of MRCP such as T2 HASTE thick slab and T2 SPACE. MRCP in the volunteers was performed before and 10 minutes after oral administration of the grape syrup solution. The concentrations of potassium, iron, and manganese in grape syrup were 34.8, 2.4 and 1 mg/l, respectively. The in vitro study showed significantly lower signal to noise ratio using the grape syrup samples comparing to the grape syrup/lemon. The effective signal suppression for the in vitro study and considerable enhancement of negative contrast in the post-contrast MRCP was obtained using a T2 HASTE thick slab sequence. As a conclusion, the grape syrup solution can be used for signal suppression of the gastrointestinal tract fluid in MRCP as an oral negative contrast material. A T2 HASTE thick slab sequence produces suitable contrast in MRCP images using the grape syrup solution.

Early stages of development of the alar fascia (human specimens at 6-12weeks of development).

In recent years, there has been much discussion concerning the cervical fasciae. The aim of this study is to confirm and to describe the development of the alar fascia as well as its relationship with nearby structures. Histological preparations of 25 human embryos (6-8weeks of development) and 25 human fetuses (9-12weeks of development) were studied bilaterally using a conventional optical microscope. Our study confirms the existence of the alar fascia and permits three stages to be established during its development. The initial stage (1st), corresponding to the 6th week of development (Carnegie stages 18-19), is characterized by the beginning of the alar fascia primordium in the retroesophageal space at the level of C7-T1. In the formation stage (2nd), corresponding to the 7th and 8th weeks of development (Carnegie stages 20-23), the alar fascia primordium grows upwards and reaches the level of C2-C3. In the maturation stage (3rd), beginning in the 9th week of development, the visceral, alar and prevertebral fasciae can be identified. The alar fascia divides the retrovisceral space (retropharyngeal and retroesophageal) into two spaces: one anterior (between the alar fascia and the visceral fascia and extending from C1 to T1, named retropharyngeal or retroesophageal space according to the level) and the other posterior (between the alar fascia and the prevertebral fascia, named danger space). We suggest that this latter space be named the retroalar space. This study suggests that alar fascia development is related to mechanical factors and that the alar fascia permits the sliding of the pharynx and the oesophagus during swallowing.

Weak Domain Models of T1 spaces

The main objective of this paper is to study some aspects of weak domains. We first show that every meet continuous weak domain is a domain. Then we prove that a dcpo P is exact iff the weakly way-below relation is the smallest approximating w-auxiliary relation. It is then shown that for each T1 space, Zhao and Xi's dcpo model is a weak algebraic domain, and hence a weak domain. As a consequence, we have that a weak algebraic domain need not be well-filtered and that every T1 space has a weak domain model, which strengthens a result of Mashburn.

Uniform Topological Spaces Based on BF-ideals in Negative Non-involutive Residuated Lattices

In this paper, the uniform topological spaces are established based on the congruences induced by BF-ideals and some of their properties are discussed in negative non-involutive residuated lattices. The following conclusions are proved: (i) every uniform topological space is first-countable, zero-dimensional, disconnected, locally compact and completely regular. (ii) a uniform topological space is a T1 space iff it is a T2 space. (iii) the lattice and adjoint operations in a negative non-involutive residuated lattice are continuous with respect to the uniform topology, which make the negative non-involutive residuated lattice to be topological negative non-involutive residuated lattice. Meanwhile, some necessary and sufficient conditions for the uniform topological spaces to be compact and discrete are obtained. The results of this paper have a positive role to reveal internal features of negative non-involutive residuated lattices at a topological level.

Induced Topologies on the Poset of Finitely Generated Saturated Sets

In [R. Heckmann, K. Keimel, Quasicontinuous Domains and the Smyth Powerdomain, Electronic Notes in Theoretical Computer Science 298 (2013), 215–232], Heckmann and Keimel proved that a dcpo P is quasicontinuous iff the poset FinP of nonempty finitely generated upper sets ordered by reverse inclusion is continuous. We generalize this result to general topological spaces in this paper. More precisely, for any T0 space (X,τ) and U∈τ, we construct a topology τF generated by the basic open subsets UF={↑F∈FinX:F⊆U}. It is shown that a T0 space (X,τ) is a hypercontinuous lattice iff τF is a completely distributive lattice. In particular, we prove that if a poset P satisfies property DINTop, then P is quasi-hypercontinuous iff FinP is hypercontinuous.

On SI2-continuous Spaces

In this paper, we construct a new way below relation from any given T0 space by making use of the cut operator, and introduce the concepts of SI2-continuous spaces and SI2-quasicontinuous spaces. The main results are: (1) a space (X,τ) is SI2-continuous iff the set X equipped with the SI2-topology τSI2 is a C-space; (2) a space (X,τ) is SI2-quasicontinuous iff (X,τSI2) is a locally hypercompact space; (3) a space is SI2-continuous iff it is a meet SI2-continuous and SI2-quasicontinuous space.

A note on the tightness of Gδ-modifications

We construct a normal countably tight T1 space X with t(Xδ)>2ω. This is an answer to the question posed by Dow-Juhász-Soukup-Szentmiklóssy-Weiss [5]. We also show that if the continuum is not so large, then the tightness of Gδ-modifications of countably tight spaces can be arbitrary large up to the least ω1-strongly compact cardinal.

On Ramsey properties, function spaces, and topological games

An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juh?sz, we note that the ?strong? version of this statement, where the second player is restricted to selecting single points rather than finite subsets, holds for all T3 spaces without isolated points. Continuing this investigation, we also consider games related to selective sequential separability, and demonstrate results analogous to those for selective separability. In particular, strong selective sequential separability in the presence of the Ramsey property may be reduced to a weaker condition on a countable sequentially dense subset. Additionally, ?- and ?-covering properties on X are shown to be equivalent to corresponding sequential properties on Cp(X). A strengthening of the Ramsey property is also introduced, which is still equivalent to ?2 and ?4 in the context of Cp(X).

Fibrewise Pairwise Soft Separation Axioms

The main idea of this research is to study fibrewise pairwise soft forms of the more important separation axioms of ordinary bitopology named fibrewise pairwise soft T0 spaces, fibrewise pairwise soft T1 spaces, fibrewise pairwise soft R0 spaces, fibrewise pairwise soft Hausdorff spaces, fibrewise pairwise soft functionally Hausdorff spaces, fibrewise pairwise soft regular spaces, fibrewise pairwise soft completely regular spaces, fibrewise pairwise soft normal spaces and fibrewise pairwise soft functionally normal spaces. In addition we offer some results concerning it.

On ultracompact spaces in ZF

We work in set theory without the Axiom of Choice (AC) and establish the following results:1.“Products of ultracompact spaces are ultracompact” + “there exists a free ultrafilter on ω” (UF(ω)) is equivalent to AC in ZFA;2.“Products of ultracompact spaces are ultracompact” does not imply AC in ZF;3.AC is equivalent to each of “products of ultracompact spaces are countably compact”, “products of ultracompact spaces are sequentially limit point compact”, “products of ℵ0-bounded spaces are countably compact”, “products of ℵ0-bounded spaces are sequentially limit point compact”, and “products of ℵ0-bounded spaces are ℵ0-bounded” in ZFA;4.UF(ω) is equivalent to each of “every ultracompact space is sequentially limit point compact” and “every ultracompact T2 space is sequentially limit point compact”;5.The Axiom of Countable Choice (ACω) is equivalent to “every ℵ0-bounded space is countably compact”;6.ACω+UF(ω) is equivalent to each of “every ultracompact space is countably compact”, “products of countably many ultracompact spaces are countably compact”, “products of countably many ultracompact spaces are sequentially limit point compact”, and “products of countably many ℵ0-bounded spaces are ℵ0-bounded”;7.Each of “a T3 space is ultracompact, if and only if, it is ℵ0-bounded” and “a Tychonoff space is ultracompact, if and only if, it is ℵ0-bounded” is equivalent to “every filter on ω can be extended to an ultrafilter on ω” (BPI(ω)), and thus each of the above statements is equivalent to the statement “the Stone space βω of all ultrafilters on ω is compact”. Thus we provide the exact characterizations of the above topological results, the former by Vaughan in “Countably compact and sequentially compact spaces”, and the latter by Bernstein in “A new kind of compactness for topological spaces”, as a specific renowned weak choice principle (which is strictly weaker than the Boolean Prime Ideal Theorem in ZF);8.ACω implies Ginsburg's theorem in “Some results on the countable compactness and pseudocompactness of hyperspaces”, namely “a T2 space X is ultracompact, if and only if, the hyperspace CL(X) (the set of non-empty closed subsets of X with the Vietoris topology) is ultracompact”, which in conjunction with UF(ω) implies the weak choice principle ACfinω (AC restricted to countable families of non-empty finite sets).

Some New Results on Soft n-T4 Spaces

Göçür and Kopuzlu showed that any soft T₄ space, may not be a soft T₂ space (also may not be a soft T₃ space). In this case, they described a new soft separation axiom which is called soft n-T₄ space. Then they indicated that any soft n-T₄ space is soft T₃ space also (Göçür and Kopuzlu, 2015b). In the present paper we showed that if (X,τ,E) is a soft n-T₄ space, topological space (X,τ_e ) is a T₄ space for all e∈ E. Then we indicated that any Soft Metric space is also soft n-T₄ space. Consequently, we indicated that any Soft Metric space ⟹ Soft n-T_4 space ⟹ Soft T_3 space ⟹ Soft T_2 space ⟹ soft T_1 space ⟹ soft T_0 space also.

Diagnostic Accuracy of High-Resolution Black-Blood MRI in the Evaluation of Intracranial Large-Vessel Arterial Occlusions

3D high-resolution black-blood MRI or MR vessel wall imaging allows evaluation of the intracranial arterial wall and extraluminal pathology. We investigated the diagnostic accuracy and reliability of black-blood MRI for the intraluminal detection of large-vessel arterial occlusions. We retrospectively identified patients with intracranial arterial occlusions, confirmed by CTA or DSA, who also underwent 3D black-blood MRI with nonenhanced and contrast-enhanced T1 sampling perfection with application-optimized contrasts by using different flip angle evolution (T1 SPACE) sequences. Black-blood MRI findings were evaluated by 2 independent and blinded neuroradiologists. Large-vessel intracranial arterial segments were graded on a 3-point scale (grades 0-2) for intraluminal baseline T1 hyperintensity and contrast enhancement. Vessel segments were considered positive for arterial occlusion if focal weak (grade 1) or strong (grade 2) T1-hyperintense signal and/or enhancement replaced the normal intraluminal black-blood signal. Thirty-one patients with 38 intracranial arterial occlusions were studied. The median time interval between black-blood MRI and CTA/DSA reference standard studies was 2 days (range, 0-20 days). Interobserver agreement was good for T1 hyperintensity (κ = 0.63) and excellent for contrast enhancement (κ = 0.89). High sensitivity (100%) and specificity (99.8%) for intracranial arterial occlusion diagnosis was observed with either intraluminal T1 hyperintensity or contrast-enhancement imaging criteria on black-blood MRI. Strong grade 2 intraluminal enhancement was maintained in >80% of occlusions irrespective of location or chronicity. Relatively increased strong grade 2 intraluminal T1 hyperintensity was noted in chronic/incidental versus acute/subacute occlusions (45.5% versus 12.5%, P = .04). Black-blood MRI with or without contrast has high diagnostic accuracy and reliability in evaluating intracranial large-vessel arterial occlusions with near-equivalency to DSA and CTA.