B is a b-open set if $B\subset Cl(IntB) \cup Int(ClB)$

A topological space $X$ is b-disconnected if it can be expressed as a union of two disjoint non-empty b-open sets. Otherwise, $X$ is said to be b-connected.

Earlier I asked on MSE about b-connected, b-disconnected as well as totally b-disconnected sets, and I found that real line provides sufficient examples of them except b-connected. But despite one really good (partial) answer, I am still looking for spaces that have non-trivial b-connected sets, and if possible, non metrizable examples too. My attempts have been futile, so now I seek from MO.

P.S. b-open sets comes from here