Let $X$ be a topological space and $A\subseteq X$. We say that $A$ is a regular closed if $A=\text{cl}(\text{int}(A))$ Is there a topological space $X$ such that $X$ isn't a discrete space and for that every closed subset of $X$ is a regular closed set? Obviusly, if $X$ is discrete then every c...

An open subset U of a space X is regular if it equals the interior of its closure, as we learn from the Wikipedia glossary of topology. Furthermore, the regular open subsets of a space (any space) form a complete Boolean algebra. I'm coming to this from logic and algebra, with not much backgroun...

Well, for starters, if your space is $T_1$ (so that one-point sets are closed), then it must be discrete. Proof: Suppose $X$ is $T_1$ and every open subset of $X$ is regular. Let $y \in X$. Then $X \backslash \{y\}$ is open. Now $X \backslash \{y\}$ must also be closed. For if not, then its ...

It turns out that every open set is regular $\iff$ every open set is clopen. (Hence, for example, a non-discrete space satisfying this condition is obtained by putting the topology $\{\varnothing,\{1\},\{2,3\},\{1,2,3\}\}$ on the set $\{1,2,3\}$.) ($\Leftarrow$) The proof in this direction is ob...

I was reading about the regular closed sets. The definition is Let $X$ be a topological space and $A\subseteq X$. We say that $A$ is a regular closed if $A=\text{cl}(\text{int}(A))$ Then, one question came to my mind: is there a topological space $X$ such that $X$ isn't a discrete space an...

Given a partition $P$ on a set $X$, we can define a topology whose open sets are unions of elements of $P$. In this topology, open sets and closed sets are the same, so all closed sets are regular closed. (If $P$ is the finest partition this is the discrete topology; if $P$ is the coarsest topo...

Let $X$ be a topological space and $A\subseteq X$. We say that $A$ is a regular closed if $A=\text{cl}(\text{int}(A))$ Is there a topological space $X$ such that $X$ isn't a discrete space and for that every closed subset of $X$ is a regular closed set? Obviusly, if $X$ is discrete then every c...

$\DeclareMathOperator\Cl{cl}\DeclareMathOperator\Int{int}\let\bez\smallsetminus$If every closed set is regular, then every closed (or open) set is clopen: given a closed set $A$, $B=A\bez\Int(A)$ is a closed set with $\Int(B)=\varnothing$, thus $B=\Cl(\Int(B))=\varnothing$, and $A$ is open. It fo...