IJPAM: Volume 83, No. 2 (2013)

A CANTOR $p-$ARY DECOMPOSITION
ON THE HILBERT CUBE

Aniruth Phon-On
Prince of Songkla University
Pattani Campus
Pattani, 94000, THAILAND
and
Centre of Excellence in Mathematics
CHE, Si Ayutthaya Rd., Bangkok, 10400, THAILAND


Abstract. Given a positive odd integer $p$ with $p\geq 3,$ the Cantor $p-$ary $C_p$ set and the Cantor $p-$ary function $f_p$ are constructed. $C_p$ is a generalization of the Cantor set in the case that the measure of the set $C_p$ is still zero and $f^{\infty}_p$ defined on the Hilbert Cube $Q$ is a generalization of the Cantor function. Also, for any $s\in (0, 1),$ let

\begin{displaymath}G^s_{f^{\infty}_{p}}=\{\{s\}\times \Big(f^{\infty}_{p}\Big)^{-1}(c)~\vert~ c\in Q_2\}\end{displaymath}

and $S$ is the set of all singletons in $\Big([0, s)\cup (s, 1]\Big)\times Q_2.$ Then $G=G^s_{f^{\infty}_{p}}\cup S$ is an upper semi continuous decomposition on the Hilbert Cube $Q.$ Moreover, $Q/G$ is homeomorphic to $Q.$

Received: September 1, 2012

AMS Subject Classification: 54C50

Key Words and Phrases: decomposition, upper semi continuous, Cantor $p-$ary set

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DOI: 10.12732/ijpam.v83i2.5 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 83
Issue: 2