Search Results (17)

If $(X,\mathcal{U})$ is a Hausdorff uniform space, we define the uniform weight $w(X,\mathcal{U})$ as the smallest cardinal $\kappa$ such that $\mathcal{U}$ has a basis of cardinality $\kappa$. An important topological cardinal of a Tychonoff space X is the number of cozero sets of X, which we denote as $z\left(X\right)$. It is known that $w(X,\mathcal{U})\le z(X\times X)$ for every compatible uniformity $\mathcal{U}$ of X. We do not know if $z(X\times X)$ can be replaced by $z\left(X\right)$. We concentrate ourselves in $w(X,{\mathcal{U}}_n)$, where ${\mathcal{U}}_n$ is the fine uniformity of X, i.e., the one having the family of normal covers as a basis. We establish upper bounds for $w(X,{\mathcal{U}}_n)$ using the character and pseudocharacter in extensions of $X\times X$ or using the cardinal $z\left(X\right)$. We also find some generalizations of the equivalence: $w(X,{\mathcal{U}}_n)={\aleph }_0$ if and only if X is metrizable and the set of non-isolated points of X is compact. Full article

In this paper, we deal with continuous multi-utility representations for closed preorders. We introduce the definition of a net-compact topology, which generalizes the concept of a sequentially compact topology. Indeed, a sequentially compact and first countable topological space is net-compact. First, we show that if every closed preorder on a net-compact Hausdorff topological space has a continuous multi-utility representation, then the topology is normal. Second, we prove that every closed preorder on a normal and net-compact Hausdorff topological space admits a continuous multi-utility representation. Full article

For the set $\mathsf{I}\left(X\right)$ of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space $\mathsf{I}\left(X\right)$ is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space ${\mathsf{I}}_{\tau }\left(X\right)$ of $\tau$-smooth idempotent probability measures on X and show that the space ${\mathsf{I}}_{\tau }\left(X\right)$ is Čech-complete if and only if the given space X is Čech-complete. Full article

The classical Rolle’s theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line. Full article

We consider the large class $\mathfrak{G}$ of locally convex spaces that includes, among others, the classes of $\left(DF\right)$-spaces and $\left(LF\right)$-spaces. For a space E in class $\mathfrak{G}$ we have characterized that a subspace Y of $(E,\sigma (E,E^{\prime }))$, endowed with the induced topology, is analytic if and only if Y has a $\sigma (E,E^{\prime })$-compact resolution and is contained in a $\sigma (E,E^{\prime })$-separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class $\mathfrak{G}$. The mentioned characterization follows from the following analogous result: The space $C\left(X\right)$ of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology $\xi$ stronger or equal than the pointwise topology ${\tau }_p$ of $C\left(X\right)$ is analytic iff $\left(C\right(X),\xi )$ is separable and is covered by a compact resolution. Full article

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups. Full article

We present some lifting theorems for continuous order-preserving functions on locally and $\sigma$-compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and $\sigma$-compact Hausdorff topological space has a continuous multi-utility representation if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. Such a characterization is also presented by introducing a lifting property of ≾-C-compatible continuous order-preserving functions on closed subspaces. The assumption of paracompactness is also used in connection to lifting conditions. Full article

The novel concept of focality is introduced for Borel probability measures on compact Hausdorff topological spaces. We characterize focal Borel probability measures as those Borel probability measures that are strictly positive on every nonempty open subset. We also prove the existence of focal Borel probability measures on compact metric spaces. Lastly, we prove that the set of focal (regular) Borel probability measures is convex but not extremal in the set of all (regular) Borel probability measures. Full article

The present paper comes across the main steps that were laid from Zadeh’s fuzziness and Atanassov’s intuitionistic fuzzy sets to Smarandache’s indeterminacy and to Molodstov’s soft sets. Two hybrid methods for assessment and decision making, respectively, under fuzzy conditions are also presented using suitable examples that use soft sets and real intervals as tools. The decision making method improves on an earlier method of Maji et al. Further, it is described how the concept of topological space, the most general category of mathematical spaces, can be extended to fuzzy structures and how to generalize the fundamental mathematical concepts of limit, continuity compactness and Hausdorff space within such kinds of structures. In particular, fuzzy and soft topological spaces are defined and examples are given to illustrate these generalizations. Full article

In this paper, we characterize spaces of $L^{\infty }$-functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work is analogous to established results concerning invariant spaces of continuous and measurable functions on a compact Hausdorff space. The case for $L^{\infty }$-functions cannot be proved in the same way when endowed with the norm-topology, but a similar argument can be used when the space of $L^{\infty }$-functions is given the weak*-topology, as we show in this paper. Full article

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in a Hausdorff topological space based on the locally compactable neighborhood systems without considering strictly (metrized) Borel variety. The null extension gives rise to a quasi sigma-semiring based on sigma-neighborhoods, which are rectifiable in view of Dieudonné measure in n-space. The concepts of symmetric signed measure, uniformly pushforward measure, and its interval-valued Lebesgue variety within a topological measure space are introduced. The symmetric signed measure preserves the total ordering on the real line; however, the collapse of symmetry admits Dieudonné measure within the topological space. The locally constant measures in compact supports in sigma-neighborhood systems are invariant under topological deformation retraction in a simply connected space where the sequence of deformation retractions induces a strongly convergent sequence of measures. Moreover, the extended sigma-structures in an automorphic and isomorphic topological space preserve the properties of uniformly pushforward measure. The Haar-measurable group algebraic structures equivalent to additive integer groups arise under the locally constant and signed measures as long as the topological space is non-compact and the null-extended sigma-neighborhood system admits compact groups. The comparative analyses of the proposed concepts with respect to existing results are outlined. Full article

The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space. The fibered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p-normed 2-spheres. The 2-quasinormed variants of p-normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p-normed 2-spheres generate finite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact fibre can prepare a homotopy loop in the fundamental group within the fibered topological (C, R) space. It is shown that the holomorphic condition is a requirement in the topological (C, R) space to preserve a convex path-component. However, the topological projections of p-normed 2-spheres on the disjoint holomorphic complex subspaces retain the path-connection property irrespective of the projective points on real subspace. The local fundamental groups of discrete-loop variety support the formation of a homotopically Hausdorff (C, R) space. Full article

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 $n_J$, where $n_J$ is the dimension of X associated with ∂. Therefore, we have ${\check{H}}_p(X;\mathbb{Z})$, where $0\le p\le n=n_J$. For a continuous self-map f on X, let $\alpha \in J$ be an open cover of X and $L_f\left(\alpha \right)=\left\{L_f\left(U\right)\right|U\in \alpha \}$. Then, there exists an open fiber cover ${\dot{L}}_f\left(\alpha \right)$ of $X^f$ induced by $L_f\left(\alpha \right)$. In this paper, we define a topological fiber entropy $en{t}_L\left(f\right)$ as the supremum of $ent(f,{\dot{L}}_f\left(\alpha \right))$ through all finite open covers of $X^f=\{L_f\left(U\right);U\subset X\}$, where $L_f\left(U\right)$ is the f-fiber of U, that is the set of images $f^n\left(U\right)$ and preimages $f^{-n}\left(U\right)$ for $n\in \mathbb{N}$. Then, we prove the conjecture $log\rho \le en{t}_L\left(f\right)$ for f being a continuous self-map on a given compact Hausdorff space X, where $\rho$ is the maximum absolute eigenvalue of $f_{*}$, which is the linear transformation associated with f on the Čech homology group ${\check{H}}_{*}(X;\mathbb{Z})={\displaystyle \underset{i=0}{\overset{n}{⨁}}}{\check{H}}_i(X;\mathbb{Z}).$ Full article

Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5). Full article

Via co-compact open sets we introduce co-$T_2$ as a new topological property. We show that this class of topological spaces strictly contains the class of Hausdorff topological spaces. Using compact sets, we characterize co-$T_2$ which forms a symmetry. We show that co-$T_2$ propoerty is preserved by continuous closed injective functions. We show that a closed subspace of a co-$T_2$ topological space is co-$T_2$. We introduce co-regularity as a weaker form of regularity, s-regularity as a stronger form of regularity and co-normality as a weaker form of normality. We obtain several characterizations, implications, and examples regarding co-regularity, s-regularity and co-normality. Moreover, we give several preservation theorems under slightly coc-continuous functions. Full article