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 $\leftrightarrow \{x\} \in \tau$


A set is open iff its complement is closed. The same holds for preopen and semiopen sets.

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. $[0, 1] ⊆ \mathbb{R}$.


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