ON THE S_{\lambda}(X) AND {\lambda}-ZERO DIMENSIONAL SPACES

Document Type : Research Note

Authors

1 Department of Science, Petroleum University of Technology, P.O. Box 6318714317, Ahvaz, Iran.

2 Department of Science, Ahvaz Islamic Azad University, P.O. Box 6134937333, Ahvaz, Iran.

Abstract

Let $S_\lambda(X)=\{f\in C(X) : |X\setminus Z(f)|<\lambda\}$, such that $\lambda$ is a regular cardinal
number with $\lambda\leq |X|$.
It is generalization of $C_F (X)=S_{\aleph_0}(X)$ and
$SC_F(X)=S_{\aleph_1}(X)$. Using
this concept we extend some of the basic results concerning the socle
to $S_\lambda(X)$. It is shown that
if $X$ is a $\lambda$-pseudo discrete space, then $C_{K,\lambda}(X)\subseteq S_{\lambda}(X)$.
$S_{\lambda}$-completely regular spaces are investigated.
Consequently, $X$ is a $S_{\aleph_1}$-completely regular space if and only if $X$ is $\aleph_1$-zero dimensional space.
$S_{\lambda}P$-spaces are introduced and studied.

Keywords

References

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Journal of Algebraic Systems
Volume 10, Issue 1
September 2022
Pages 179-188

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