Generalizations of sobriety of quantale-valued cotopological spaces

Abstract

In this paper, we extend bounded sobriety and k-bounded sobriety to the setting of quantale-valued cotopological spaces (short for Q-cotopological spaces). The main results are: (1) The category BSobQ-CTop of all bounded sober Q-cotopological spaces is a full reflective subcategory of the category SQ-CTop of all stratied Q-cotopological spaces; (2) We present the relationships among Hausdor, T1, sobriety, bounded sobriety and k-bounded sobriety in the setting of Q-cotopological spaces; (3) For a linearly ordered quantale Q, a topological space X is bounded (resp., k-bounded) sober if and only if the corresponding Q-cotopological space ω_{Q}(X) is bounded (resp., k-bounded) sober, where ω_{Q} : Top → SQ-CTop is the well-known Lowen functor in fuzzy topology.