proper

Definitions

1. adjective

   Suited or acceptable to the purpose or circumstances; fit, suitable.

   Examples

   • Use the proper tool for the job.
   • She wore proper attire for the interview.
   ≈correct≈right≈apt≈prudent≈upright +52 more
2. adjective

   Following the established standards of behavior or manners; correct or decorous.

   Examples

   • He always uses proper table manners.
   • She spoke in a proper, formal tone.
   ≈appropriate≈decent≈good≈polite≈right +104 more
3. adjective

   Used to designate a particular person, place, or thing. Proper nouns are usually written with an initial capital letter.

   Examples

   • "Paris" and "Anna" are both proper nouns.
   • The teacher explained the difference between common and proper nouns.
4. adjective

   Pertaining exclusively to a specific thing or person; particular.

   Examples

   • Each species has its proper habitat.
   • The word has a meaning proper to legal contexts.
   ≈individual≈singular≈concrete≈discrete≈individual +5 more
5. adjective

   In the strict sense; within the strict definition or core (of a specified place, taxonomic order, idea, etc).

   Examples

   • He lives in the city proper, not the suburbs.
   • The museum's collection proper excludes the gift shop items.
6. adjective

   Such that the preimage of every compact set is compact.

   Examples

   • The map is proper because compact sets pull back to compact sets.
   • Mathematicians verified the function was proper before applying the theorem.
7. adjective

   Continuous, mapping closed sets to closed sets, and such that the preimage of every point is compact.

   Examples

   • The projection is a proper map in this topology.
   • They proved the map was proper by checking each fiber.
8. adjective

   Separated, of finite type, and universally closed.

   Examples

   • The scheme was shown to be proper over the base field.
   • Properness is a key condition in the theorem's proof.
9. adjective

   Such that unique morphism from the variety to k is proper (as above).

   Examples

   • The variety is proper if this morphism satisfies the condition.
   • Researchers checked whether the variety was proper over the field.
10. adjective

    Such that every closed ball is compact.

    Examples

    • The space is proper since every closed ball is compact.
    • Properness of the metric space simplified the proof.