Related papers: Arzel\`a-Ascoli theorem in uniform spaces

In the paper, we recall the Wallman compactification of a Tychonoff space $T$ (denoted by $\text{Wall}(T)$) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between…

We generalize the Arzel\`a-Ascoli theorem in the space of continuous maps on a compact interval with values in Euclidean N-space by providing a quantitative link between the Hausdorff measure of noncompactness in this space and a natural…

We generalize the Arzel\`a-Ascoli theorem to the setting of matrix order unit spaces, extending the work of Antonescu-Christensen on unital C*-algebras. This gives an affirmative answer to a question of Antonescu and Christensen.

As indicated in the title, this paper was inspired by Smithson's work on Arzel\`a-Ascoli theorem. Up to this point, the author's interest in the topic has led him to generalize Arzel\`a-Ascoli theorem in the context of uniform spaces. This…

In this paper we investigate Arzela Ascoli Theorem in quasi cone metric space, which is a generalization of metric space. We prove some interesting results using forward and backward toplologies, forward and backward continuity and forward…

A Tychonoff space $X$ is called ({\em sequentially}) {\em Ascoli} if every compact subset (resp. convergent sequence) of $C_k(X)$ is equicontinuous, where $C_k(X)$ denotes the space of all real-valued continuous functions on $X$ endowed…

The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing…

The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the $T_{3 \frac{1}{2}}$ space. Motivated by the generalized Arzel\`a-Ascoli theorem for Tichonoff space $T$ via Wallman…

In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural…

Following [3] we say that a Tychonoff space $X$ is an Ascoli space if every compact subset $\mathcal{K}$ of $C_k(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_\mathbb{R}$-space, hence any…

We prove that, in the space of all probabilistic continuous functions from a probabilistic metric space G to the set $\Delta$ + of all cumulative distribution functions vanishing at 0, the space of all 1-Lipschitz functions is compact if…

In what follows, essentially two things will be accomplished: Firstly, it will be proven that a version of the Arzel\`a--Ascoli theorem and the Fr\'echet--Kolmogorov theorem are equivalent to the axiom of countable choice for subsets of…

A version of Arzel\`a-Ascoli theorem for $X$ being $\sigma$-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Hammerstein and Urysohn operators. Two fixed point theorems, for Hammerstein and…

A topological space $X$ is said to be an Ascoli space if any compact subset $K$ of $C_k(X)$ is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the $k_R$-property and the Ascoli property of…

The proof by Ullmo and Zhang of Bogomolov's conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for ``small points'' in the associated complex abelian variety. We study the…

We show how one can obtain solutions to the Arzel\`a-Ascoli theorem using suitable applications of the Bolzano-Weierstra{\ss} principle. With this, we can apply the results from \cite{aK} and obtain a classification of the strength of…

In this article we combine the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. We define new spaces of functions on which study the equations, prove a version of…

The paper deals with Ascoli spaces $C_p(X)$ and $C_k(X)$ over Tychonoff spaces $X$. The class of Ascoli spaces $X$, i.e. spaces $X$ for which any compact subset $K$ of $C_k(X)$ is evenly continuous, essentially includes the class of…

The classical Arzela-Ascoli theorem is a compactness result for families of functions depending on bounds on the derivatives of the functions, and is of invaluable use in many fields of mathemathics. In this paper, inspired by a result of…

We generalize the dual notions of "expansion" and "collapse" so they can be applied to arbitrary metric spaces. We also expand the theory to allow for infinitely many such moves. Those tools are then employed to prove a variety of…