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 Distance to Spaces of Semicontinuous and Continuous Functions

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Carlos Angosto
Keyword(s):  
Topological Space ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Lower Semicontinuous Functions ◽  
Upper Semicontinuous Functions ◽  
Semicontinuous Functions

Given a topological spaceX, we establish formulas to compute the distance from a functionf∈RXto the spaces of upper semicontinuous functions and lower semicontinuous functions. For this, we introduce an index of upper semioscillation and lower semioscillation. We also establish new formulas about distances to some subspaces of continuous functions that generalize some classical results.

Quasicontinuous functions, minimal usco maps and topology of pointwise convergence

2010 ◽  
Vol 60 (4) ◽  
Author(s):  
Dušan Holý ◽  
Ladislav Matejíčka
Keyword(s):  
Pointwise Convergence ◽  
Rocky Mountain ◽  
Set Valued Mapping ◽  
Upper Semicontinuous Functions ◽  
Quasicontinuous Functions ◽  
Minimal Usco Map ◽  
Pointwise Limit ◽  
Topology Of Pointwise Convergence ◽  
Semicontinuous Functions

AbstractIn [HOLÁ, Ľ.—HOLÝ, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces, Rocky Mountain J. Math.] a complete answer is given, for a Baire space X, to the question of when the pointwise limit of a sequence of real-valued quasicontinuous functions defined on X is quasicontinuous. In [HOLÁ, Ľ.—HOLÝ, D.: Minimal USCO maps, densely continuous forms and upper semicontinuous functions, Rocky Mountain J. Math. 39 (2009), 545–562], a characterization of minimal USCO maps by quasicontinuous and subcontinuous selections is proved. Continuing these results, we study closed and compact subsets of the space of quasicontinuous functions and minimal USCO maps equipped with the topology of pointwise convergence. We also study conditions under which the closure of the graph of a set-valued mapping which is the pointwise limit of a net of set-valued mappings, is a minimal USCO map.

scholarly journals A unified approach to continuous and certain non-continuous functions II

1990 ◽  
Vol 41 (1) ◽  
pp. 57-74 ◽  
Author(s):  
J.K. Kohli
Keyword(s):  
Functional Analysis ◽  
Continuous Functions ◽  
Unified Theory ◽  
Unified Approach ◽  
Lower Upper ◽  
Upper Semicontinuous ◽  
Weakly Continuous ◽  
Upper Semicontinuous Functions ◽  
Almost Continuous ◽  
Semicontinuous Functions

A unified theory of continuous and certain non-continuous functions, initiated in an earlier paper, is further elaborated. The proposed theory provides a common platform for dealing simultaneously with continuous functions and a host of non-continuous functions including lower (upper) semicontinuous functions, almost continuous functions, weakly continuous functions (encountered in functional analysis), c-continuous functions, δ-continuous functions, semiconnected functions, H-continuous functions s-continuous functions, ε-continuous functions of Klee and several other variants of continuity.

scholarly journals Density topology and pointwise convergence

2003 ◽  
Vol 4 (2) ◽  
pp. 509 ◽  
Author(s):  
Wladyslaw Wilczynski
Keyword(s):  
Pointwise Convergence ◽  
Continuous Functions ◽  
Density Topology ◽  
Topology Of Pointwise Convergence

<p>We shall show that the space of all approximately continuous functions with the topology of pointwise convergence is not homeomorphic to its category analogue.</p>

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