We study a topology on a space of functions, called sticking topology, with the property to be the weakest among the topologies preserving continuity. In suitable frameworks, this topology preserves borelianity, local integrability, right continuity and other properties. It is coarser than the locally uniform convergence and it allows the presence of gliding humps as we show on examples. We prove relative compactness criteria for this topology and we consider some extensions.
Let E be a topological space and F a uniform space. We introduce a new topology (in fact a uniform structure) called the V-congergence on the space of applications from E to F such that C(E,F) is closed for this topology and the restriction of this topology to C(E,F) is equivalent to pointwise convergence. In other words this topology is the coarsest preserving continuity. We give a criterion of convergence for this topology not involving the limit. Among properties preserved are mesurability and alpha-borelianity for a countable ordinal alpha.
The separately continuity topology is considered and some its properties are investigated. With help of these properties a generalization of Sierpinski theorem on determination of real separately continuous function by its values on an arbitrary dense set is obtained.
The most general definition of a continuous function requires that the preimage of any open set be open. Thus, to discuss continuity in the abstract, it is necessary to first define a topology, which tells us which sets in a space are open. Such a topology is given by three axioms that describe how the open sets in a topology behave. In this paper we shall consider a change to one of these axioms and determine how this change affects the continuity of a function.
Groups with a topology that is in consistent one way or another with the algebraic structure are considered. Classical groups with a topology are topological, paratopological, semitopological, and quasitopological groups. We also study other ways of matching topology and algebraic structure. The minimum requirement in this paper is that the group is a right semitopological group (such groups are often called right topological groups). We study when a group with a topology is a topological group; research in this direction began with the work of Deane Montgomery and Robert Ellis. (Invariant) se…
For a $T_0$ space $X$, let $\mk (X)$ be the poset of all compact saturated sets of $X$ with the reverse inclusion order. The space $X$ is said to have property Q if for any $K_1, K_2\in \mk (X)$, $K_2\ll K_1$ in $\mk (X)$ if{}f $K_2\subseteq \ii~\!K_1$. In this paper, we give several connections among the well-filteredness of $X$, the sobriety of $X$, the local compactness of $X$, the core compactness of $X$, the property Q of $X$, the coincidence of the upper Vietoris topology and Scott topology on $\mk (X)$, and the continuity of $x\mapsto\ua x : X \longrightarrow Σ~\!\! \mk (X)$ (where $Σ~\…
This chapter examines how positivity and order play out in two important questions in mathematical economics, and in so doing, subjects the postulates of continuity, additivity and monotonicity to closer scrutiny. Two sets of results are offered: the first departs from Eilenberg's (1941) necessary and sufficient conditions on the topology under which an anti-symmetric, complete, transitive and continuous binary relation exists on a topologically connected space; and the second, from DeGroot's (1970) result concerning an additivity postulate that ensures a complete binary relation on a σ-algebr…
In a metric space, such as the real numbers with their standard metric, a set A is open if and only if no sequence with terms outside of A has a limit inside A. Moreover, a metric space is compact if and only if every sequence has a converging subsequence. However, in a general topological space these equivalences may fail. Unfortunately this fact is sometimes overlooked in introductory courses on general topology, leaving many students with misconceptions, e.g. that compactness would always be equal to sequence compactness. The aim of this article is to show how sequences might fail to charac…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as well as their applications to real numbers. Various methods to construct topological spaces are presented.
Niagara University（尼亚加拉大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）；Fairfield University（费尔菲尔德大学）
We study connectivity preserving multivalued functions between digital images. This notion generalizes that of continuous multivalued functions studied mostly in the setting of the digital plane $Z^2$. We show that connectivity preserving multivalued functions, like continuous multivalued functions, are appropriate models for digital morpholological operations. Connectivity preservation, unlike continuity, is preserved by compositions, and generalizes easily to higher dimensions and arbitrary adjacency relations.
We propose a generalization of continuous lattices and domains through the concept of enriched closure space, defined as a closure space equipped with a preclosure operator satisfying some compatibility conditions. In this framework we are able to define a notion of way-below relation; an appropriate definition of continuity then naturally follows. Characterizations of continuity of the enriched closure space and necessary and sufficient conditions for the interpolation property are proved. We also draw a link between continuity and the possibility for the subsets that are open with respect to…
Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space L(H) of linear bounded operators on H with weak operator topology. We prove that if U is a measurable map from G to L(H) then it is continuous. This result was known before for separable H. To prove this, we generalize a known theorem on nonmeasuralbe unions of point finite families of null sets. We prove also that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous. This relies, in turn, …
In this paper we focus on the set-open topologies on the group $\mathcal{H}(X)$ of all self-homeomorphisms of a topological space $X$ which yield continuity of both the group operations, product and inverse function. As a consequence, we make the more general case of Dijkstra's theorem. In this case a homogeneously encircling family $\mathcal{B}$ consists of regular open sets and the closure of every set from $\mathcal{B}$ is contained in the finite union of connected sets from $\mathcal{B}$. Also we proved that the zero-cozero topology of $\mathcal{H}(X)$ is the relativisation to $\mathcal{H}…
We consider two relatively natural topologizations of the set of all cellular automata on a fixed alphabet. The first turns out to be rather pathological, in that the countable space becomes neither first-countable nor sequential. Also, reversible automata form a closed set, while surjective ones are dense. The second topology, which is induced by a metric, is studied in more detail. Continuity of composition (under certain restrictions) and inversion, as well as closedness of the set of surjective automata, are proved, and some counterexamples are given. We then generalize this space, in the …
