Neutrosophic Simply b-Open Set in Neutrosophic Topological Spaces

In this paper, we procure the notions of neutrosophic simply b-open set, neutrosophic simply b-open cover, and neutrosophic simply b-compactness via neutrosophic topological spaces. Then, we establish some remarks, propositions, and theorems on neutrosophic simply b-compactness. Further, we furnish some counter examples where the result fails.

Topological Manifolds and Polish Spaces

We give,as a preliminary result, some topological characterizations of locally compact second-countable Hausdorff spaces. Then we show that a topological manifold, with boundary or not,is precisely a Polish space with a coordinate open cover; this connects geometry with descriptive set theory.

Elementary Proof of Existence of a Subordinate Smooth Partition of Unity on Smooth Manifolds

We give in particular an elementary proof of the existence of a smooth partition of unity subordinate to any given open cover for smooth manifolds. As a side note, given are also two elementary proofs of the existence of a subordinate partition of unity for topological manifolds. These in particular fill a gap in the related literature.

The Topological Entropy Conjecture

For a compact Hausdorff space X, let J be the ordered set associated with the set of all finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with ∂. Therefore, we have Hˇp(X;Z), where 0≤p≤n=nJ. For a continuous self-map f on X, let α∈J be an open cover of X and Lf(α)={Lf(U)|U∈α}. Then, there exists an open fiber cover L˙f(α) of Xf induced by Lf(α). In this paper, we define a topological fiber entropy entL(f) as the supremum of ent(f,L˙f(α)) through all finite open covers of Xf={Lf(U);U⊂X}, where Lf(U) is the f-fiber of U, that is the set of images fn(U) and preimages f−n(U) for n∈N. Then, we prove the conjecture logρ≤entL(f) for f being a continuous self-map on a given compact Hausdorff space X, where ρ is the maximum absolute eigenvalue of f*, which is the linear transformation associated with f on the Čech homology group Hˇ*(X;Z)=⨁i=0nHˇi(X;Z).

A-paracompactness and Strongly A-screenability in Topological Groups

A space is said to be strongly A-screenable if there exists a σ-discrete refinement for each open cover. In this article, we have investigated some of the features of A-paracompact and strongly A-screenable spaces in topological and semi topological groups. We predominantly show that (i) Topological direct product of (countably) A-paracompact topological group and a compact topological group is (countably) A-paracompact topological group. (ii) All the left and right cosets of a strongly A-screenable subset H of a semi topological group (G, ∗, τ ) are strongly A-creenable.

Onpairwise Compactness in Fuzzy Finebi-Topological Spaces

In this paper the concepts of Pairwise ( , ) Tif Tjf fuzzy fine dually open cover, Pairwise ( , ) Tif Tjf fuzzy fine dually compact spacesand Pairwise ( , ) Tif Tjf fuzzy fine connected spaces are introduced and also an equivalent statement on Pairwise ( , ) Tif Tjf fuzzy fine dually compact spaces is established.

Sufficient conditions for expansive group action

Existence of expansivity for group action [Formula: see text] depends on algebraic properties of [Formula: see text] and the topology of [Formula: see text]. We give an expansive action of a solvable group on [Formula: see text] while there is no expansive action of a solvable group on a dendrite [Formula: see text]. We prove that a continuous action [Formula: see text] on a compact metric space [Formula: see text] is expansive if and only if there exists an open cover [Formula: see text] such that for any other open cover [Formula: see text], [Formula: see text] for some finite set [Formula: see text]. In this paper, we introduce the notion of topological expansivity of a group action [Formula: see text] on a [Formula: see text]-paracompact space [Formula: see text]. If a [Formula: see text]-paracompact space [Formula: see text] admits topologically expansive action, then [Formula: see text] is Hausdorff space. We also show that a continuous action [Formula: see text] of a finitely generated group [Formula: see text] on a compact Hausdorff uniform space [Formula: see text] is expansive with an expansive neighborhood [Formula: see text] if and only if for every [Formula: see text] there is an entourage [Formula: see text] such that for every two [Formula: see text]-pseudo orbit [Formula: see text] if [Formula: see text] for all [Formula: see text], then [Formula: see text] for all [Formula: see text]. Finally, we introduce measure [Formula: see text]-expansive actions on a uniform space. The set of all [Formula: see text]-expansive measures with common expansive neighborhood for a group action [Formula: see text] is a convex, closed and [Formula: see text]-invariant subset of the set of all Borel probability measures on [Formula: see text]. Also, we show that a group action [Formula: see text] is expansive if all Borel probability measures are [Formula: see text]-expansive or all Dirac measures [Formula: see text], [Formula: see text], have a common expansive neighborhood.

On the mathematical and foundational significance of the uncountable

We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. [Formula: see text] as “almost finite”, while the latter allows one to treat uncountable sets like e.g. [Formula: see text] as “almost countable”. This reduction of the uncountable to the finite/countable turns out to have a considerable logical and computational cost: we show that the aforementioned lemmas, and many related theorems, are extremely hard to prove, while the associated sub-covers are extremely hard to compute. Indeed, in terms of the standard scale (based on comprehension axioms), a proof of these lemmas requires at least the full extent of second-order arithmetic, a system originating from Hilbert–Bernays’ Grundlagen der Mathematik. This observation has far-reaching implications for the Grundlagen’s spiritual successor, the program of Reverse Mathematics, and the associated Gödel hierarchy. We also show that the Cousin lemma is essential for the development of the gauge integral, a generalization of the Lebesgue and improper Riemann integrals that also uniquely provides a direct formalization of Feynman’s path integral.

Local unstable entropies of partially hyperbolic diffeomorphisms

Consider a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$. Following the ideas in establishing the local variational principle for topological dynamical systems, we introduce the notions of local unstable metric entropies (and local unstable topological entropy) relative to a Borel cover ${\mathcal{U}}$ of $M$. It is shown that they coincide with the unstable metric entropy (and unstable topological entropy, respectively), when ${\mathcal{U}}$ is an open cover with small diameter. We also define the unstable tail entropy in the sense of Bowen and the unstable topological conditional entropy in the sense of Misiurewicz, and demonstrate that both of them vanish. Some generalizations of these results to the case of unstable pressure are also investigated.

Testing for homologies in the axial skeleton of primitive echinoderms

The extraxial axial theory is used to investigate homology of ambulacral and oral plating because it predicts terminal branching and terminal addition of plates in the axial skeleton, although exceptions to the former may occur in some Paleozoic echinoderms. The variety of morphological designs and anomalous individuals also provide tests of plate homology. Homology of ambulacra is generally accepted, with the hydropore and/or single gonopore in Carpenter’s CD interray. In the 2-1-2 ambulacral pattern the unbranched ambulacrum is always in Carpenter’s A ray. All ambulacral morphology requires just three instructions: ‘grow,’ ‘branch,’ and ‘stop.’ The range of variation in echinoderms with fewer than five ambulacra implies that both the ‘branch’ and ‘stop’ instructions acted independently in all five rays. Numbers of ambulacra may or may not correlate with numbers of orals. Two basic patterns of ‘cystoid’ oral plating occur; with a single radial (circum-oral, CO) plate from each ambulacrum plus a sixth in the CD interray, and with all six interradial peri-oral (PO) plates, with two in the CD interambulacrum. Five ‘orals’ may involve loss of PO3 or PO6. Erect ambulacral structures are lost first in taphonomy and so poorly known. All ambulacral skeletal elements bear the same topological relationship to ambulacral soft tissues. Where branched ambulacra occur, the trunk or flooring plates are often modified first brachiolars or pinnulars. Both brachioles and pinnules may arise from facets developed on one or two flooring plates. Terminal addition of plates, spacing of brachioles/pinnules, and lack of musculature to open cover plates all suggest that ‘cystoids’ had extensions of the water vascular system in their ambulacra.