Embeddings of Bishop spaces

2020 ◽  
Vol 30 (1) ◽  
pp. 349-379
Author(s):  
Iosif Petrakis
Keyword(s):  
Classical Theory ◽  
Extension Theorem ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Constructive Approach ◽  
Point Function ◽  
Constructive Theory ◽  
Zero Sets ◽  
Informal System ◽  
Inductive Definitions

Abstract We develop the basic constructive theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of topological spaces. The theory of Bishop spaces is a constructive approach to point-function topology and a natural constructive alternative to the classical theory of the rings of continuous functions. Our most significant result is the translation of the classical Urysohn extension theorem within the theory of Bishop spaces. The related theory of the zero sets of a Bishop topology is also included. We work within $\textrm{BISH}^{\ast }$, Bishop’s informal system of constructive mathematics $\textrm{BISH}$ equipped with inductive definitions with rules of countably many premises.

Some Smoothness Properties of Measures on Topological Spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 165-172
Author(s):  
Shankar Hegde
Keyword(s):  
Continuous Functions ◽  
Topological Spaces ◽  
Bounded Linear ◽  
Natural Classification ◽  
Zero Sets ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Bounded Linear Functional ◽  
Uniformly Closed

AbstrctV. S. Varadarajan has classified the bounded linear functional on the algebra C(X) of bounded continuous functions on a topological space X, according to the properties of their smoothness and related this classification to the corresponding natural classification of finitely additive regular measures on the zero sets of X. In this paper, some of these results are extended to the linear functionals on an arbitrary uniformly closed algebra A of bounded functions on a set X.

scholarly journals Some strong and weak form of z-continuity via Cl*

2019 ◽  
Vol 27 (1) ◽  
pp. 85-101
Author(s):  
A. R. Prasannan ◽  
J. Biswas
Keyword(s):  
Weak Form ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Pragmatic Approach ◽  
Zero Sets ◽  
New Class ◽  
Class Of Functions

Abstract This paper mainly dedicated on overview of zero sets in Ideal topological spaces. We also introduce a new class of functions which generalizes the class of continuous functions and investigate its position in the hierarchy of continuous functions on Ideal topological spaces. Moreover, these new sets (zero*-ℐ-set) which is a pragmatic approach to characterize completely Hausdorff spaces.

scholarly journals R-continuous functions

1992 ◽  
Vol 15 (1) ◽  
pp. 57-64 ◽  
Author(s):  
Ch. Konstadilaki-Savvapoulou ◽  
D. Janković
Keyword(s):  
Continuous Functions ◽  
Strong Form ◽  
Topological Spaces ◽  
Strong Continuity ◽  
Closed Graph ◽  
Continuity Properties

A strong form of continuity of functions between topological spaces is introduced and studied. It is shown that in many known results, especially closed graph theorems, functions under consideration areR-continuous. Several results in the literature concerning strong continuity properties are generalized and/or improved.

scholarly journals On Softβ-Open Sets and Softβ-Continuous Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Metin Akdag ◽  
Alkan Ozkan
Keyword(s):  
Continuous Functions ◽  
Soft Set ◽  
Topological Spaces

We introduce the concepts softβ-interior and softβ-closure of a soft set in soft topological spaces. We also study softβ-continuous functions and discuss their relations with soft continuous and other weaker forms of soft continuous functions.

Ultrafilter-completeness on zero-sets of uniformly continuous functions

2019 ◽  
Vol 252 ◽  
pp. 27-41
Author(s):  
Asylbek A. Chekeev ◽  
Tumar J. Kasymova
Keyword(s):  
Continuous Functions ◽  
Zero Sets ◽  
Uniformly Continuous

scholarly journals A note on weakly θ-continuous functions

1989 ◽  
Vol 12 (1) ◽  
pp. 9-13 ◽  
Author(s):  
M. Mrševic ◽  
I. L. Reilly
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
New Class ◽  
Weakly Continuous ◽  
Class Of Functions

Recently a new class of functions between topological spaces, called weaklyθ-continuous functions, has been introduced and studied. In this paper we show how an appropriate change of topology on the domain of a weaklyθ-continuous function reduces it to a weakly continuous function. This paper examines some of the consequences of this result.

scholarly journals Adelic descent theory

2017 ◽  
Vol 153 (8) ◽  
pp. 1706-1746
Author(s):  
Michael Groechenig
Keyword(s):  
Vector Bundles ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Reductive Algebraic Group ◽  
Perfect Complexes ◽  
Descent Theory ◽  
Closed Fields ◽  
Ring Of Continuous Functions ◽  
Reconstruction Theorem ◽  
Algebraically Closed Fields

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

scholarly journals ‎INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO ‎‎COMPARABLE CONTRA-$\alpha-$CONTINUOUS (CONTRA-$C-$CONTINUOUS) FUNCTIONS

2019 ◽  
pp. 13
Author(s):  
Majid Mirmiran ◽  
Binesh Naderi
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

‎A necessary and sufficient condition in terms of lower cut sets ‎are given for the insertion of a contra-continuous function ‎between two comparable real-valued functions on such topological ‎spaces that kernel of sets are open‎. 

The Black–Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral

2021 ◽  
Vol 15 (01) ◽  
pp. 45-59
Author(s):  
E. M. Bonotto ◽  
M. Federson ◽  
P. Muldowney
Keyword(s):  
Extension Theorem ◽  
Continuous Functions ◽  
Sample Space ◽  
Asset Valuation ◽  
Itô Calculus ◽  
Black Scholes ◽  
Pricing Theory ◽  
Simple Sets ◽  
Random Times ◽  
Riemann Integration

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka[Formula: see text] formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

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