Open Access

## A. Vadivel, Shiventhira Devi Sathaananthan, S. Tamilselvan and G. Saravanakumar

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In this paper we introduce the concepts of pre generalized double fuzzy $\mathcal{Z}$-open, generalized double fuzzy $\mathcal{Z}$-extremely is connected spaces. Also we introduce the concepts of generalized double fuzzy $\mathcal{Z}$-basically disconnected space and related sets such as $(r,\kappa)$-generalized fuzzy $\mathcal{Z}$ open-$F_{\sigma}$, $(r,\kappa)$ generalized fuzzy $\mathcal{Z}$closed-$G_{\delta}$ sets and maps such as Generalized double fuzzy $\mathcal{Z}F_\sigma$-open, $\mathcal{Z}G_\delta$ continuous and $\mathcal{Z}F_\sigma$-irresolute function. Some characterizations of the concepts are studied.

Keywords:
$pgDF\mathcal{Z}O$, $gDF\mathcal{Z}ed$, $(r,\kappa)$-$gf\mathcal{Z}o$- F_{\sigma}$,$(r,\kappa)$-$gf\mathcal{Z}o$-$G_{\delta}$,$gDF\mathcal{Z}b\$ disconnected space.