Network character and tightness of the compact-open topology

Richard N. Ball; Anthony W. Hager

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 3, page 473-482
  • ISSN: 0010-2628

Abstract

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For Tychonoff X and α an infinite cardinal, let α def X : = the minimum number of α  cozero-sets of the Čech-Stone compactification which intersect to X (generalizing ℝ -defect), and let rt X : = min α max ( α , α def X ) . Give C ( X ) the compact-open topology. It is shown that τ C ( X ) ≤ n χ C ( X ) ≤ rt X = max ( L ( X ) , L ( X ) def X ) , where: τ is tightness; n χ is the network character; L ( X ) is the Lindel"of number. For example, it follows that, for X Čech-complete, τ C ( X ) = L ( X ) . The (apparently new) cardinal functions n χ C and rt are compared with several others.

How to cite

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Ball, Richard N., and Hager, Anthony W.. "Network character and tightness of the compact-open topology." Commentationes Mathematicae Universitatis Carolinae 47.3 (2006): 473-482. <http://eudml.org/doc/249890>.

@article{Ball2006,
abstract = {For Tychonoff $X$ and $\alpha $ an infinite cardinal, let $\alpha \operatorname\{def\} X := $ the minimum number of $\alpha $ cozero-sets of the Čech-Stone compactification which intersect to $X$ (generalizing $\mathbb \{R\}$-defect), and let $\operatorname\{rt\} X := \min _\alpha \max (\alpha , \alpha \operatorname\{def\} X)$. Give $C(X)$ the compact-open topology. It is shown that $\tau C(X)\le n\chi C(X) \le \operatorname\{rt\}X=\max (L(X),L(X) \operatorname\{def\} X)$, where: $\tau $ is tightness; $n\chi $ is the network character; $L(X)$ is the Lindel"of number. For example, it follows that, for $X$ Čech-complete, $\tau C(X)=L(X)$. The (apparently new) cardinal functions $n\chi C$ and $\operatorname\{rt\}$ are compared with several others.},
author = {Ball, Richard N., Hager, Anthony W.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {compact-open topology; network character; tightness; defect; Lindelöf number; tightness; defect; Lindelöf number},
language = {eng},
number = {3},
pages = {473-482},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Network character and tightness of the compact-open topology},
url = {http://eudml.org/doc/249890},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Ball, Richard N.
AU - Hager, Anthony W.
TI - Network character and tightness of the compact-open topology
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 3
SP - 473
EP - 482
AB - For Tychonoff $X$ and $\alpha $ an infinite cardinal, let $\alpha \operatorname{def} X := $ the minimum number of $\alpha $ cozero-sets of the Čech-Stone compactification which intersect to $X$ (generalizing $\mathbb {R}$-defect), and let $\operatorname{rt} X := \min _\alpha \max (\alpha , \alpha \operatorname{def} X)$. Give $C(X)$ the compact-open topology. It is shown that $\tau C(X)\le n\chi C(X) \le \operatorname{rt}X=\max (L(X),L(X) \operatorname{def} X)$, where: $\tau $ is tightness; $n\chi $ is the network character; $L(X)$ is the Lindel"of number. For example, it follows that, for $X$ Čech-complete, $\tau C(X)=L(X)$. The (apparently new) cardinal functions $n\chi C$ and $\operatorname{rt}$ are compared with several others.
LA - eng
KW - compact-open topology; network character; tightness; defect; Lindelöf number; tightness; defect; Lindelöf number
UR - http://eudml.org/doc/249890
ER -

References

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  8. McCoy R., Function spaces which are k -spaces, Topology Proc. 5 (1980), 139-154. (1980) MR0624467
  9. McCoy R., Ntantu I., Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics 1315, Springer, Berlin, 1988. Zbl0647.54001MR0953314
  10. Mrowka S., On E -compact spaces. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 597-605. (1966) Zbl0161.19603MR0206896
  11. Mrowka S., ℛ -compact spaces with weight X < E x p ℛ X , Proc. Amer. Math. Soc. 128 (2000), 3701-3709. (2000) MR1690997

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