Topologizing Different Classes of Real Functions

1994 ◽  
Vol 46 (06) ◽  
pp. 1188-1207 ◽  
Author(s):  
Krzysztof Ciesielski
Keyword(s):  
Continuum Hypothesis ◽  
Continuous Functions ◽  
Affirmative Answer ◽  
Linear Functions ◽  
Hausdorff Topology ◽  
Completely Regular ◽  
Differentiable Functions ◽  
Real Functions ◽  
Baire One ◽  
Number Of Classes

AbstractThe purpose of this paper is to examine which classesof functions fromcan be topologized in a sense that there exist topologies τ1and τ2onandrespectively, such thatis equal to the class C(τ1, τ2) of all continuous functions. We will show that the Generalized Continuum Hypothesis GCH implies the positive answer for this question for a large number of classes of functionsfor which the sets {x : f(x) = g(x)} are small in some sense for all f, g ∈f ≠ g. The topologies will be Hausdorff and connected. It will be also shown that in some model of set theory ZFC with GCH these topologies could be completely regular and Baire. One of the corollaries of this theorem is that GCH implies the existence of a connected Hausdorff topology T onsuch that the class L of all linear functions g(x) = ax + b coincides with. This gives an affirmative answer to a question of Sam Nadler. The above corollary remains true for the classof all polynomials, the classof all analytic functions and the class of all harmonic functions.We will also prove that several other classes of real functions cannot be topologized. This includes the classes of C∞functions, differentiable functions, Darboux functions and derivatives.

Separate approximate continuity implies measurability

1973 ◽  
Vol 73 (3) ◽  
pp. 461-465 ◽  
Author(s):  
Roy O. Davies
Keyword(s):  
Continuum Hypothesis ◽  
Measurable Cardinal ◽  
Continuous Functions ◽  
Affirmative Answer ◽  
Baire Class ◽  
Continuous Real Function ◽  
Approximate Continuity ◽  
Class 1 ◽  
Measure Spaces ◽  
First Baire Class

It is known that a real-valued function f of two real variables which is continuous in each variable separately need not be continuous in (x, y), but must be in the first Baire class (1). Moreover if f is continuous in x for each y and merely measurable in y for each x then it must be Lebesgue-measurable (7), and this result can be extended to more general product spaces (2). However, the continuum hypothesis implies that this result fails if continuity is replaced by approximate continuity, as can be seen from the proof of Theorem 2 of (2). This makes Mišik's question (5) very natural: is a function which is separately approximately continuous in both variables necessarily Lebesgue-measurable? Our main aim is to establish an affirmative answer. It will be shown that such a function must in fact be in the second Baire class, although not necessarily in the first Baire class (unlike approximately continuous functions of one variable (3)). Finally, we show that the existence of a measurable cardinal would imply that a separately continuous real function on a product of two topological finite complete measure spaces need not be product-measurable.

Methods of comparison of families of real functions in porosity terms

2019 ◽  
Vol 26 (4) ◽  
pp. 643-654 ◽  
Author(s):  
Stanisław Kowalczyk ◽  
Małgorzata Turowska
Keyword(s):  
Uniform Convergence ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
Absolutely Continuous ◽  
Real Functions ◽  
Upper Continuous ◽  
Baire One ◽  
Lower Continuous

Abstract We consider some families of real functions endowed with the metric of uniform convergence. In the main results of our work we present two methods of comparison of families of real functions in porosity terms. The first method is very general and may be applied to any family of real functions. The second one is more convenient but can be used only in the case of path continuous functions. We apply the obtained results to compare in terms of porosity the following families of functions: continuous, absolutely continuous, Baire one, Darboux, also functions of bounded variation and porouscontinuous, ρ-upper continuous, ρ-lower continuous functions.

scholarly journals Preimages of Residual Sets of Continuous Functions under Operation of Superposition

2005 ◽  
Vol 12 (4) ◽  
pp. 763-768
Author(s):  
Artur Wachowicz
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Real Functions ◽  
Second Category ◽  
Residual Sets

Abstract Let 𝐶 = 𝐶[0, 1] denote the Banach space of continuous real functions on [0, 1] with the sup norm and let 𝐶* denote the topological subspace of 𝐶 consisting of functions with values in [0, 1]. We investigate the preimages of residual sets in 𝐶 under the operation of superposition Φ : 𝐶 × 𝐶* → 𝐶, Φ(𝑓, 𝑔) = 𝑓 ○ 𝑔. Their behaviour can be different. We show examples when the preimages of residual sets are nonresidual of second category, or even nowhere dense, and examples when the preimages of nontrivial residual sets are residual.

Real Functions, Covers and Bornologies

2021 ◽  
Vol 78 (1) ◽  
pp. 199-214
Author(s):  
Lev Bukovský
Keyword(s):  
Topological Space ◽  
Pointwise Convergence ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Upper Semicontinuous Functions ◽  
Covering Properties ◽  
Real Functions ◽  
Topology Of Pointwise Convergence ◽  
Semicontinuous Functions

Abstract The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.

scholarly journals Equilibria and Systemic Risk in Saturated Networks

2021 ◽  
Author(s):  
Leonardo Massai ◽  
Giacomo Como ◽  
Fabio Fagnani
Keyword(s):  
Systemic Risk ◽  
Nash Equilibria ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Linear Functions ◽  
Shock Propagation ◽  
Fundamental Study ◽  
Network Games ◽  
Bifurcation Phenomenon

We undertake a fundamental study of network equilibria modeled as solutions of fixed-point equations for monotone linear functions with saturation nonlinearities. The considered model extends one originally proposed to study systemic risk in networks of financial institutions interconnected by mutual obligations. It is one of the simplest continuous models accounting for shock propagation phenomena and cascading failure effects. This model also characterizes Nash equilibria of constrained quadratic network games with strategic complementarities. We first derive explicit expressions for network equilibria and prove necessary and sufficient conditions for their uniqueness, encompassing and generalizing results available in the literature. Then, we study jump discontinuities of the network equilibria when the exogenous flows cross certain regions of measure 0 representable as graphs of continuous functions. Finally, we discuss some implications of our results in the two main motivating applications. In financial networks, this bifurcation phenomenon is responsible for how small shocks in the assets of a few nodes can trigger major aggregate losses to the system and cause the default of several agents. In constrained quadratic network games, it induces a blow-up behavior of the sensitivity of Nash equilibria with respect to the individual benefits.

Strict Topology on Spaces of Continuous Vector-Valued Functions

1979 ◽  
Vol 31 (4) ◽  
pp. 890-896 ◽  
Author(s):  
Seki A. Choo
Keyword(s):  
Tensor Product ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Convex Space ◽  
Completely Regular ◽  
Vector Valued Functions ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Locally Convex

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

CONTINUOUS PIECEWISE LINEAR FUNCTIONS

2005 ◽  
Vol 10 (1) ◽  
pp. 77-99 ◽  
Author(s):  
CHARALAMBOS D. ALIPRANTIS ◽  
DAVID HARRIS ◽  
RABEE TOURKY
Keyword(s):  
Piecewise Linear ◽  
Continuous Functions ◽  
Linear Functions ◽  
Dimensional Euclidean Space ◽  
Piecewise Linear Functions ◽  
Space Of Continuous Functions ◽  
One Dimensional ◽  
Mathematical Background ◽  
Linear Regressions ◽  
Special Case

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on them-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean formwhereX=(X1,X2, …,Xm) is some random vector, {Ej}j∈Jis a finite family of finite sets.

scholarly journals Lebesgue Measurability of Separately Continuous Functions and Separability

2007 ◽  
Vol 2007 ◽  
pp. 1-4
Author(s):  
V. V. Mykhaylyuk
Keyword(s):  
Topological Space ◽  
Chain Condition ◽  
Continuous Functions ◽  
Negative Answer ◽  
Regular Space ◽  
Property A ◽  
Completely Regular ◽  
Open Set ◽  
Countable Chain Condition ◽  
Countable Chain

A connection between the separability and the countable chain condition of spaces withL-property (a topological spaceXhasL-property if for every topological spaceY, separately continuous functionf:X×Y→ℝand open setI⊆ℝ,the setf−1(I)is anFσ-set) is studied. We show that every completely regular Baire space with theL-property and the countable chain condition is separable and constructs a nonseparable completely regular space with theL-property and the countable chain condition. This gives a negative answer to a question of M. Burke.

scholarly journals Some results involving weak compactness in C(X), C(υX) and C(X)′

1975 ◽  
Vol 19 (3) ◽  
pp. 221-229 ◽  
Author(s):  
I. Tweddle
Keyword(s):  
Dual Space ◽  
Present Note ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Regular Space ◽  
Compact Sets ◽  
Completely Regular ◽  
Countably Compact ◽  
Countable Compactness

The main aim of the present note is to compare C(X) and C(υX), the spaces of real-valued continuous functions on a completely regular space X and its real 1–1 compactification υX, with regard to weak compactness and weak countable compactness. In a sense to be made precise below, it is shown that C(X) and C(υX) have the same absolutely convex weakly countably compact sets. In certain circumstances countable compactness may be replaced by compactness, in which case one obtains a nice representation of the Mackey completion of the dual space of C(X) (Theorems 5, 6, 7).

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