semienclosed

Semienclosed is not a standard or widely agreed-upon term in mainstream topology. Because the literature does not settle on a single definition, the meaning of semienclosed can vary by author or context. In many discussions related to semi-open and semi-closed sets, semienclosed appears as a nonstandard or informal label for a property that sits between open/closed notions or as a synonym for one of the established weak notions.

Related concepts commonly discussed in topology include semi-open and semiclosed sets. A subset A of a topological space X is called semi-open if A is contained in the closure of its interior, that is, A ⊆ cl(int(A)). A set is sometimes called semiclosed if int(cl(A)) ⊆ A ⊆ cl(A). These notions form part of a broader family of “weak” separation properties that generalize openness and closedness.

Because semienclosed lacks a universal definition, some authors may use it to mean a set that is contained in the closure of an open set (A ⊆ cl(U) for some open U), while others may intend it as a synonym for semiclosed in the sense above, or as a descriptive label for a set that satisfies a condition like int(cl(A)) ⊆ A ⊆ cl(A). In examples such as the real line with the standard topology, intervals like (0,1] illustrate how these loose notions can apply to familiar sets.

In practice, readers should consult the specific text or author to determine what semienclosed means in that context. Without a standard definition, the term remains ambiguous and potentially interchangeable with related weak closure/open properties.