A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

Novel Idea of Gδ-α-Locally Continuous Functions

The new concepts of a neutrosophic Gδ set and neutrosophic Gδ-α-locally closed sets are introduced. Also, a neutrosophic εGδ-α-locally quasi neighborhood, neutrosophic Gδ-α-locally continuous function, neutrosophic Gδ-α-local T2 space, neutrosophic Gδ-α-local Urysohn space, neutrosophic Gδ-α-local connected space, and neutrosophic Gδ-α-local compact space are discussed and some interesting properties are established.

SUPRA CO BT URYSOHN SPACE IN SUPRA TOPOLOGICAL SPACES

In this article, the author interpolate prolific notion of co $bT^{\mu}$ graph and co $bT^{\mu}$ urysohn space by utilizing the concept of bT$^{\mu}$ set connected functions in supra topological spaces.

On spaces defined by Pytkeev networks

The notions of networks and k-networks for topological spaces have played an important role in general topology. Pytkeev networks, strict Pytkeev networks and cn-networks for topological spaces are defined by T. Banakh, and S. Gabriyelyan and J. K?kol, respectively. In this paper, we discuss the relationship among certain Pytkeev networks, strict Pytkeev networks, cn-networks and k-networks in a topological space, and detect their operational properties. It is proved that every point-countable Pytkeev network for a topological space is a quasi-k-network, and every topological space with a point-countable cn-network is a meta-Lindel?f D-space, which give an affirmative answer to the following problem [25, 29]: Is every Fr?chet-Urysohn space with a pointcountable cs'-network a meta-Lindel?f space? Some mapping theorems on the spaces with certain Pytkeev networks are established and it is showed that (strict) Pytkeev networks are preserved by closed mappings and finite-to-one pseudo-open mappings, and cn-networks are preserved by pseudo-open mappings, in particular, spaces with a point-countable Pytkeev network are preserved by closed mappings.

A Universal Separable Diversity

The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

CONSEQUENCES OF THE EXISTENCE OF AMPLE GENERICS AND AUTOMORPHISM GROUPS OF HOMOGENEOUS METRIC STRUCTURES

We define a simple criterion for a homogeneous, complete metric structure X that implies that the automorphism group Aut(X) satisfies all the main consequences of the existence of ample generics: it has the automatic continuity property, the small index property, and uncountable cofinality for nonopen subgroups. Then we verify it for the Urysohn space $$, the Lebesgue probability measure algebra MALG, and the Hilbert space $\ell _2 $, thus proving that Iso($$), Aut(MALG), $U\left( {\ell _2 } \right)$, and $O\left( {\ell _2 } \right)$ share these properties. We also formulate a condition for X which implies that every homomorphism of Aut(X) into a separable group K with a left-invariant, complete metric, is trivial, and we verify it for $$, and $\ell _2 $.

Effective metric model theory

This paper is a further investigation of a project carried out in Didehvar and Ghasemloo (2009) to study effective aspects of the metric logic. We prove an effective version of the omitting types theorem. We also present some concrete computable constructions showing that both the separable atomless probability algebra and the rational Urysohn space are computable metric structures.

Addendum to ‘The Katětov construction modified for a T0-quasi-metric space’

It is known that if K is a compact subset of the (separable complete) metric Urysohn space (${\mathbb U}$, d) and f is a Katětov function on the subspace K of (${\mathbb U}$, d), then there is z ∈ ${\mathbb U}$ such that d(z, x) = f(x) for all x ∈ K.Answering a question of Normann, we show in this article that the supseparable bicomplete q-universal ultrahomogeneous T0-quasi-metric space (q${\mathbb U}$, D) recently discussed by the authors satisfies a similar property for Katětov function pairs on subsets that are compact in the associated metric space (q${\mathbb U}$, Ds).