Proximity space

Key takeaways

Proximity space belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Proximity space to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Proximity space from memory before moving on to harder problems.

Reference excerpt

In topology, a proximity space, also called a nearness space, is an axiomatization of the intuitive notion of "nearness" that applies set-to-set, as opposed to the better-known point-to-set notion that characterizes topological spaces. The concept was described by Frigyes Riesz (1909) but ignored at the time. It was rediscovered and axiomatized by V. A. Efremovič in 1934 under the name of infinitesimal space, but not published until 1951. In the interim, A. D. Wallace (1941) discovered a version of the same concept under the name of separation space.

Definition A proximity space ( X , δ ) {\displaystyle (X,\delta )} is a set X {\displaystyle X} with a relation δ {\displaystyle \delta } between subsets of X {\displaystyle X} satisfying the following properties: For all subsets A , B , C ⊆ X {\displaystyle A,B,C\subseteq X}

A δ B {\displaystyle A\;\delta \;B} implies B δ A {\displaystyle B\;\delta \;A}

A δ B {\displaystyle A\;\delta \;B} implies A ≠ ∅ {\displaystyle A\neq \varnothing }

A ∩ B ≠ ∅ {\displaystyle A\cap B\neq \varnothing } implies A δ B {\displaystyle A\;\delta \;B}

A δ ( B ∪ C ) {\displaystyle A\;\delta \;(B\cup C)} if and only if ( A δ B {\displaystyle A\;\delta \;B} or A δ C {\displaystyle A\;\delta \;C} ) (For all E , {\displaystyle E,} A δ E {\displaystyle A\;\delta \;E} or B δ ( X ∖ E ) {\displaystyle B\;\delta \;(X\setminus E)} ) implies A δ B {\displaystyle A\;\delta \;B}

Proximity without the first axiom is called quasi-proximity (but then Axioms 2 and 4 must be stated in a two-sided fashion). If A δ B {\displaystyle A\;\delta \;B} we say A {\displaystyle A} is near B {\displaystyle B} or A {\displaystyle A} and B {\displaystyle B} are proximal; otherwise we say A {\displaystyle A} and B {\displaystyle B} are apart. We say B {\displaystyle B} is a proximal- or δ {\displaystyle \delta } -neighborhood of A , {\displaystyle A,} written A ≪ B , {\displaystyle A\ll B,} if and only if A {\displaystyle A} and X ∖ B {\displaystyle X\setminus B} are apart. The main properties of this set neighborhood relation, listed below, provide an alternative axiomatic characterization of proximity spaces. For all subsets A , B , C , D ⊆ X {\displaystyle A,B,C,D\subseteq X}

X ≪ X {\displaystyle X\ll X}

A ≪ B {\displaystyle A\ll B} implies A ⊆ B {\displaystyle A\subseteq B}

A ⊆ B ≪ C ⊆ D {\displaystyle A\subseteq B\ll C\subseteq D} implies A ≪ D {\displaystyle A\ll D}

( A ≪ B {\displaystyle A\ll B} and A ≪ C {\displaystyle A\ll C} ) implies A ≪ B ∩ C {\displaystyle A\ll B\cap C}

A ≪ B {\displaystyle A\ll B} implies X ∖ B ≪ X ∖ A {\displaystyle X\setminus B\ll X\setminus A}

A ≪ B {\displaystyle A\ll B} implies that there exists some E {\displaystyle E} such that A ≪ E ≪ B . {\displaystyle A\ll E\ll B.}

A proximity space is called separated if { x } δ { y } {\displaystyle \{x\}\;\delta \;\{y\}} implies x = y . {\displaystyle x=y.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Proximity space

Start with the simplest possible case. Write down what Proximity space claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Proximity space before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Proximity space ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Proximity space

In research: Proximity space appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Proximity space in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Proximity space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Closure operators, General topology, so understanding it makes those chapters shorter.
In everyday life: Look for Proximity space outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Proximity space in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Proximity space means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Proximity space out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Proximity space in simple terms?

In topology, a proximity space, also called a nearness space, is an axiomatization of the intuitive notion of "nearness" that applies set-to-set, as opposed to the better-known point-to-set notion that characterizes topological spaces. The concept was described by Frigyes Riesz (1909) but ignored a…

Why does Proximity space matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Proximity space?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Proximity space.