topologieswithout

Topologieswithout is a term used to describe a family of topologies on a fixed set X that avoid a prescribed collection of subsets, called forbidden opens. Formally, given a set X and a family F of subsets of X, the collection Topologieswithout(X,F) consists of all topologies T on X such that T does not contain any member of F; in symbols, T is a topology on X and T ∩ F = ∅. The forbidden family F should not include the empty set or the whole space X if one intends for at least one topology to exist.

Existence and basic properties: If F contains ∅ or X, Topologieswithout(X,F) is empty. When F does not contain ∅ or X, at least the indiscrete topology {∅, X} typically lies in Topologieswithout(X,F). The set Topologieswithout(X,F) is a subfamily of the full lattice of topologies on X and is closed under finite intersections, since the intersection of topologies that avoid F is again a topology that avoids F. It is not generally closed under arbitrary unions; the join (the smallest topology containing two given topologies) may introduce a forbidden open set from F.

Examples: On X = {0,1} and F = {{0}}, the indiscrete topology is in Topologieswithout, while the Sierpinski topologies and the discrete topology may be excluded if they contain {0}. More broadly, for any nonempty F not containing ∅ or X, there exist topologies avoiding F, though the specific members of Topologieswithout(X,F) depend on F and X.

Relation to broader concepts: Topologieswithout is a combinatorial constraint on the open set lattice, related to, but distinct from, subspace, quotient, or specialization concepts. It is not a standard widely used term in conventional references, but describes a natural way to study topologies under forbidden-open restrictions.