# Relation between open, semi-open and preopen

Does any one of the semi,pre and just open given by definitions below imply another one under no assumption about the space?

If $(X, \tau)$ is a topological sapce and $A \subset X$, then $A$ is called:

peropen if $A \subset Int(\overline{A})$

preclosed if $\overline{Int(A)} \subset A$

semi-open if $A \subset \overline{Int(A)}$

semi-closed if $Int(\overline{A}) \subset A$

This is the only result I know of. It is only about semi-open singletons equivalent to open singletons:

Let $x \in (X, \tau)$, then $\{x\}$ is semi-open $\leftrightarrow \{x\} \in \tau$

An open set is both preopen and semiopen, but the converse is not true. Let $D := \{(x, y) ∈ \mathbb{R}^2: x^2 + y^2 < 1\}$. The set $\{(x, y): (x, y) ∈ D, x < 0\} ∪ \{(0, 0\} ∪ \{(x, y): (x, y) ∈ D, x > 0\}$ is both preopen and semiopen but not open in $\mathbb{R}^2$.
Every dense set is preopen, but if its complement is also dense, then it is not semiopen, not semiclosed, and not preclosed. E.g. $\mathbb{Q} ⊆ \mathbb{R}$.
Any closure of and open set is semiopen, but if it's not is not clopen, then is it not preopen. E.g. $[0, 1] ⊆ \mathbb{R}$.