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2007/2008 On the Graph Convergence of Sequences of Functions
Zbigniew Grande
Real Anal. Exchange 33(2): 365-374 (2007/2008).

Abstract

Let $(X,T_X)$ and $(Y,T_Y)$ be topological spaces. A sequence $(f_n)$ of functions $f_n:X \to Y$ is graph convergent to $f:X\to Y$ if for each set $U \in T_X\times T_Y$ containing the graph $Gr(f)$ of $f$ there is an index $k$ such that $Gr(f_n) \subset U$ for $n >> k$. It is proved that if $(X,T_X)$ is a $T_1$ space, then the graph convergence implies the pointwise convergence. Moreover the uniform and graph convergences are compared, and the graph limits of sequences of continuous (quasicontinuous, cliquish, almost continuous or Darboux) functions are investigated.

Citation

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Zbigniew Grande. "On the Graph Convergence of Sequences of Functions." Real Anal. Exchange 33 (2) 365 - 374, 2007/2008.

Information

Published: 2007/2008
First available in Project Euclid: 18 December 2008

zbMATH: 1161.26002
MathSciNet: MR2458253

Subjects:
Primary: 26A15

Keywords: almost continuity , Cliquishness , continuity , Darboux property , graph convergence , Quasicontinuity , uniform and pointwise convergence

Rights: Copyright © 2007 Michigan State University Press

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Vol.33 • No. 2 • 2007/2008
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