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Regular $$L$$-fuzzy topological spaces and their topological modifications. (English) Zbl 0951.54009
Let $$L$$ be a continuous lattice with its Scott topology. There is a functor $$\omega_L$$ from the category Top of topological spaces to the category $$L$$-Top of $$L$$-topological spaces. $$\omega_L$$ sends every topological space $$(X,{\mathcal T})$$ to $$(X,\omega_L({\mathcal T}))$$, where $$\omega_L ({\mathcal T})$$ denotes the collection of all continuous functions from $$X$$ to $$L$$. Conversely, there is also a functor $$\iota_L$$ from $$L$$-Top to Top, $$\iota_L$$ sends every $$L$$-topological space $$(X,\Delta)$$ to $$(X,\iota_L (\Delta))$$, where $$\iota_L(\Delta)$$ is the coarsest topology on $$X$$ making all $$\lambda \in \Delta$$ continuous. In this paper it is showed that both of the functors $$\omega_L$$ and $$\iota_L$$ preserve regularity. Precisely, if $$(X,{\mathcal T})$$ is regular then $$(X,\omega_L ({\mathcal T}))$$ is $$L$$-regular (i.e., regular in the Hutton-Reilly sense); if $$(X,\Delta)$$ is $$L$$-regular then $$(X,\iota_L (\Delta))$$ is regular. Moreover, the relationship of $$L$$-regular spaces to $$H$$-Lindelöf spaces and $$L$$-normal spaces is discussed.

##### MSC:
 54A40 Fuzzy topology
##### Keywords:
$$L$$-regularity; Lowen functors; $$L$$-regular spaces
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