Is a circle a Jordan curve?

Yes, a perfect circle is a classic example of a Jordan curve, as it is a simple, closed, non-self-intersecting loop.

Why is it called a 'Jordan' curve?

It is named after the French mathematician Camille Jordan (1838-1922), who stated and attempted to prove the Jordan curve theorem.

Can a Jordan curve have corners?

Yes. A Jordan curve must be continuous, but it does not have to be smooth. A square, for example, is a Jordan curve made of four line segments.

Is the Jordan curve theorem obvious?

Intuitively it seems obvious, but providing a rigorous proof for all possible simple closed curves is surprisingly difficult and was a major achievement in topology.

jordan curve

Rare

UK/ˌdʒɔːdən ˈkɜːv/US/ˌdʒɔːrdən ˈkɜːrv/

Technical / Academic

My Flashcards

Definition

Meaning

In mathematics, specifically topology, a non-self-intersecting continuous loop in the plane.

A simple closed curve that divides the plane into two distinct, complementary regions: a bounded interior and an unbounded exterior. It serves as a foundational concept in the Jordan curve theorem.

Linguistics

Semantic Notes

A strictly defined term in mathematical topology; not a general descriptive phrase. Its meaning is precise and does not vary with context within mathematics.

Dialectal Variation

British vs American Usage

Differences

No significant differences; term is identical in UK and US mathematical literature. Spelling follows standard regional conventions for surrounding text (e.g., "centre" vs. "center").

Connotations

Purely technical; carries no cultural or regional connotations.

Frequency

Exclusively used in advanced mathematics (especially topology, geometry, and analysis). Frequency is essentially zero in general language and identical between UK and US in technical contexts.

Vocabulary

Collocations

strong

Jordan curve theoremsimple closedinterior of aexterior of acontinuousnon-self-intersecting

medium

planetopologicalproofboundaryregion

weak

mathematicalcomplexanalysisdefineconcept

Grammar

Valency Patterns

[determiner] + Jordan curve + [verb] (e.g., 'A Jordan curve divides...')the Jordan curve + [prepositional phrase] (e.g., 'the Jordan curve theorem')

Vocabulary

Synonyms

Strong

simple loop

Neutral

simple closed curve

Weak

continuous closed curvenon-self-intersecting loop

Vocabulary

Antonyms

self-intersecting curveopen curveline segment

Phrases

Idioms & Phrases

“None. It is a technical term.”

Usage

Context Usage

Business

Never used.

Academic

Used exclusively in advanced mathematics, specifically in topology, complex analysis, and geometry.

Everyday

Virtually never used; unknown to the general public.

Technical

Core term in mathematical topology. Used in proofs, definitions, and theoretical discussions.

Examples

By Part of Speech

verb

British English

The path can be approximated by a Jordan curve.
One must first construct a Jordan curve to proceed with the proof.

American English

The proof requires us to Jordan curve the boundary.
You cannot simply Jordan curve an arbitrary set of points.

adverb

British English

The region is Jordan-curve bounded.
The path behaves Jordan-curve-like near the singularity.

American English

The region is Jordan curve bounded.
The path behaves in a Jordan curve-like manner.

adjective

British English

The Jordan-curve property is fundamental.
We studied the Jordan-curve approach to the problem.

American English

The Jordan curve property is fundamental.
We studied the Jordan curve approach to the problem.

Examples

By CEFR Level

(Not applicable for this highly technical term at A2 level.)

(Not applicable for this highly technical term at B1 level.)

A Jordan curve is a basic concept in advanced geometry.
The drawing showed a loop called a Jordan curve.

The Jordan curve theorem states that any simple closed curve divides the plane into an interior and an exterior.
Analysing the properties of a Jordan curve is crucial for understanding planar topology.

Learning

Memory Aids

Mnemonic

Think of drawing a circle without lifting your pen and without crossing your own line—that's the idea of a 'simple closed' Jordan curve.

Conceptual Metaphor

A fence that perfectly separates a finite garden (interior) from the infinite outside world (exterior).

Watch out

Common Pitfalls

Translation Traps (for Russian speakers)

Do not translate 'Jordan' as the country (Иордания). It is a name. The standard Russian translation is 'жорданова кривая' (zhordanova krivaya).
The word 'curve' is singular, even though 'кривая' in Russian is a feminine singular adjective-noun pair.

Common Mistakes

Using it as a general term for any curve (it is a specific topological concept).
Capitalising 'curve' (only 'Jordan' is capitalised as it is an eponym).
Pronouncing 'Jordan' as in the country (the 'or' is typically as in 'for', not 'ore').

Practice

Quiz

Fill in the gap

The famous theorem states that any simple closed curve in the plane divides it into two regions.

Multiple Choice

What is a Jordan curve?

FAQ

Frequently Asked Questions

: Yes, a perfect circle is a classic example of a Jordan curve, as it is a simple, closed, non-self-intersecting loop.
: It is named after the French mathematician Camille Jordan (1838-1922), who stated and attempted to prove the Jordan curve theorem.
: Yes. A Jordan curve must be continuous, but it does not have to be smooth. A square, for example, is a Jordan curve made of four line segments.
: Intuitively it seems obvious, but providing a rigorous proof for all possible simple closed curves is surprisingly difficult and was a major achievement in topology.