closureProperty

P78666

predicate

Indicates that applying a specified operation to elements of a set always produces an element that is still within that same set.

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All labels observed (1)

Label	Occurrences
closureProperty canonical	39

Description generation (PDg)

The one-sentence description above was generated by prompting gpt-5.1 with the predicate name and this instruction.

Instruction

Given a predicate that represents a relationship or action between entities, generate a one-sentence description explaining its meaning.  
# Instructions
Focus on describing the relationship, not the entities themselves. 
# Response Format
Begin the description with \' Indicates...\'

Input

Predicate: closureProperty

Generated description

Indicates that applying a specified operation to elements of a set always produces an element that is still within that same set.

Sample triples (39)

Subject	Object
NP-completeness	closed under polynomial-time many-one reductions ⓘ
Hausdorff surface form: Hausdorff space	subspaces of Hausdorff spaces are Hausdorff ⓘ
Hausdorff surface form: Hausdorff space	finite products of Hausdorff spaces are Hausdorff ⓘ
Hausdorff surface form: Hausdorff space	arbitrary products of Hausdorff spaces are Hausdorff ⓘ
Hausdorff surface form: Hausdorff space	closed subspaces of Hausdorff spaces are Hausdorff ⓘ
Hadamard product (of power series)	set of all power series over a field is closed under Hadamard product ⓘ
NP-hardness	closed under polynomial-time reductions ⓘ
Karp reductions surface form: Karp reduction	transitive ⓘ
Kleene star	preserves regularity of languages ⓘ
Kleene star	preserves context-freeness of languages ⓘ
Dedekind domain	localization at multiplicative sets yields Dedekind domains (under suitable conditions) ⓘ
Dedekind domain	finite integral extensions of Dedekind domains are Dedekind domains (under suitable hypotheses) ⓘ
NFA	closed under union ⓘ
NFA	closed under concatenation ⓘ
NFA	closed under Kleene star ⓘ
NFA	closed under intersection ⓘ
NFA	closed under complement ⓘ
Liouville numbers surface form: Liouville number	sum of a Liouville number and a rational number is a Liouville number ⓘ
Liouville numbers surface form: Liouville number	product of a nonzero rational number and a Liouville number is a Liouville number ⓘ
Liouville numbers surface form: Liouville number	set of Liouville numbers is closed under addition with rationals ⓘ
Liouville numbers surface form: Liouville number	set of Liouville numbers is closed under multiplication by nonzero rationals ⓘ
Pauli group	closed under matrix multiplication ⓘ
Pauli group	closed under taking inverses ⓘ
complexity class BPP surface form: BPP	closed under complement ⓘ
complexity class BPP surface form: BPP	closed under union ⓘ
complexity class BPP surface form: BPP	closed under intersection ⓘ
complexity class BPP surface form: BPP	closed under polynomial-time many-one reductions ⓘ
complexity class RP surface form: RP	closed under union with deterministic polynomial-time languages ⓘ
complexity class RP surface form: RP	closed under intersection with deterministic polynomial-time languages ⓘ
complexity class RP surface form: RP	closed under polynomial-time many-one reductions ⓘ
Borel set	closed under relative complementation within the σ-algebra ⓘ
Rabin automaton	closed under union ⓘ
Rabin automaton	closed under intersection ⓘ
Rabin automaton	closed under complementation ⓘ
T1 separation axiom	The closure of a singleton {x} is {x} itself. ⓘ
Kolmogorov space (T0 space) surface form: Kolmogorov space	subspaces of T0 spaces are T0 ⓘ
Kolmogorov space (T0 space) surface form: Kolmogorov space	finite products of T0 spaces are T0 ⓘ
Kolmogorov space (T0 space) surface form: Kolmogorov space	arbitrary products of T0 spaces are T0 ⓘ
Kolmogorov space (T0 space) surface form: Kolmogorov space	quotients of T0 spaces need not be T0 ⓘ