# Examples of b-connected sets?

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

Notice that if a set $$S$$ is a regular-closed subset of the space $$X$$ (that is, if $$S$$ is the closure of an open subset of $$X$$), then $$S$$ is b-open. Furthermore, any open subset is b-open. Therefore, if $$X$$ has a non-empty proper regular-closed set $$S$$, then $$S$$ and $$X \setminus S$$ shows that $$X$$ is b-disconnected. In particular, every Hausdorff space having at least two points, say $$p$$ and $$q$$, is b-disconnected. (Take $$S$$ to be the closure of a neighborhood of $$p$$ such that $$q \not\in S$$.)
To get an example of a b-connected space, let $$X$$ be the set $$\omega$$ of natural numbers with the topology consisting of sets of the form $$A_n = \{0, 1, \cdots n\}$$ along with the empty set and $$X$$. (Here $$\omega$$ can be replaced with any ordinal $$\lambda \geq 1$$; if $$\lambda = 2$$, this is Sierpinski's two-point space.) Notice that any non-empty open subset of $$X$$ contains $$0$$ and that the closure in $$X$$ of the non-empty subset $$S$$ is $$\{m, m+1, \cdots\}$$ where $$m$$ is the smallest element of $$S$$. It follows that a non-empty subset $$S$$ of $$X$$ is b-open if and only $$0 \in S$$, and, in particular, there are no two disjoint non-empty b-open subsets of $$X$$.