The least cardinal κ such that for every subset A of a space and every point x in the closure of A there exists a subset B⊆A with |B|≤κ and x in the closure of B; this invariant measures how closures are controlled by small subsets and is denoted t(X) or T(X).

Principle

Describes how tightly limit points depend on small subsubsets: a small tightness means closures are determined by small pieces, providing a bridge between pointwise closure behavior and cardinality constraints.

Demonstration

In first-countable spaces the tightness is ℵ0 because closure at a point is determined by sequences (countable sets); for a space with the cofinite topology on an uncountable set the tightness is 1 since any point in the closure of A either belongs to A or every cofinite open contains it, so a singleton suffices.

Misapplication

Assuming low tightness implies metrizability or first-countability is incorrect; tightness is only one of several conditions and does not imply the existence of countable local bases by itself.

Consequence

A bound on tightness yields cardinality controls for closures and influences hereditary properties: spaces of small tightness often allow reduction arguments replacing large sets by small witnesses when checking closure-related statements.

Reversal

Reversing tightness gives the concept of spaces where closures can require arbitrarily large subsets to witness membership of a point in the closure — high tightness indicates weak local control by small sets.

Boundary

Applies to arbitrary topological spaces and pertains to closures relative to arbitrary subsets; it does not directly prescribe the type of sequences or nets needed, and it is distinct from notions like sequentiality or Fréchet–Urysohn property.

Semantic Tension

Tightness is close to sequential order and the Fréchet–Urysohn property but differs: sequential spaces relate closures to sequences specifically, whereas tightness quantifies the cardinal size of witnesses without prescribing their form.

Synthesis

Tightness measures the minimal cardinal size of witnesses needed to determine closures at points: it captures how closure membership can be reduced to small subfamilies and interacts with other local and hereditary topological properties.