Examples of b-connected sets?

Question

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

Answer 1

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 $X$.