The extent E(X) of a topological space X is the supremum of the cardinalities of closed discrete subspaces of X; equivalently, the largest cardinal κ such that X contains a closed discrete subset of cardinality κ.

Principle

Extent measures how large a discrete configuration can be while remaining closed in the ambient space; it controls the maximal size of isolated closed pieces.

Demonstration

A discrete closed space of cardinal κ has E(X)=κ. In many separable metric spaces every closed discrete subspace is countable, so E(X)=ℵ0 for those spaces.

Misapplication

Mistaking extent for spread (which allows discrete subspaces that need not be closed) or for density (which is minimal dense size) leads to wrong conclusions; treating non-closed discrete sets as contributing to extent is incorrect.

Consequence

Extent bounds the possible sizes of closed discrete configurations and affects compactness, Lindelöf, and separation arguments; small extent can preclude large closed discrete obstructions to compactness-like properties.

Reversal

Replacing 'closed discrete' by 'arbitrary discrete' yields spread; considering minimal closed discrete sizes (infima) gives different invariants with opposite qualitative behaviour.

Boundary

Applies to topological spaces and is confined to closed discrete subspaces; it excludes discrete subspaces that are not closed and says nothing directly about open-set families or networks.

Semantic Tension

Extent is often compared with spread: both measure discrete configurations, but extent requires closedness. There is tension with compactness-related invariants since extent constrains closed discrete obstructions to compactness.

Synthesis

Extent E(X) is the supremum of cardinalities of closed discrete subsets of X; it quantifies the maximal size of discrete collections that are topologically closed and so measures closed isolated structure inside the space.