proximal neighborhood


Let X be a set and P(X) its power setMathworldPlanetmath. Let be a binary relationMathworldPlanetmath on P(X) satisfying the

following conditions, for any A,BX:

  1. 1.

    XX,

  2. 2.

    AB implies AB,

  3. 3.

    AB and CD imply ACBD,

  4. 4.

    AB implies BA ( is the complementPlanetmathPlanetmath operator)

  5. 5.

    ABCD, then AD, and

  6. 6.

    if AB, then there is CX, such that ACB.

By 1 and 4, it is easy to see that . Also, 3 and 4 show that ACBD whenever AB and CD. So is a topogenous order, which means is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath and anti-symmetric. Under this order relation, we say that B is a proximal neighborhood of A if AB.

The reason why we call B a “proximal” neighborhoodMathworldPlanetmathPlanetmath is due to the following:

Theorem 1.

Let X be a set. The following are true.

  • Let be defined as above. Define a new relation δ on P(X): AδB iff AB. Then δ so defined is a proximity relation, turning X into a proximity space.

  • Conversely, let (X,δ) is a proximity space. Define a new relation on P(X): AB iff AδB. Then satisfies the six properties above.

Proof.

Suppose first that X and are defined as above. We will verify the individual nearness relation axioms of δ by proving their contrapositives in each case, except the last axiom:

  1. 1.

    if AδB, then AB, or AB, so AB=;

  2. 2.

    suppose either A= or B=. In either case, AB, which means AδB;

  3. 3.

    if AδB, then AB, so B′′A, or BA, or BδA;

  4. 4.

    if A1δB and A2δB, then A1B and A2B, so (A1A2)B, or (A1A2)δB;

  5. 5.

    if AδB, then AB. So there is DX with AD and DB. Let C=D. Then AC and CB, or AδC and CδB.

Next, suppose (X,δ) is a proximity space. We now verify the six properties of above.

  1. 1.

    since Xδ, X, or XX;

  2. 2.

    suppose AδB, then if xA, we have xδB, implying xB=, or xB;

  3. 3.

    if AB and CD, then AδB and CδD, which means Aδ(BD) and Cδ(BD), which together imply (AC)δ(BD), or (AC)δ(BD), or ACBD;

  4. 4.

    if AB, then AδB, so BδA (as δ is symmetric, so is its complement), which is the same as BδA′′, or BA;

  5. 5.

    if AδD, then BδC (since AB and DC), so BC, a contradictionMathworldPlanetmathPlanetmath;

  6. 6.

    if AB, then AδB, so there is DX with AδD and DδB. Define C=D, then AC and CB, as desired.

This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof. ∎

Because of the above, we see that a proximity space can be equivalently defined using the proximal neighborhood concept. To emphasize its relationship with δ, a proximal neighborhood is also called a δ-neighbhorhood.

Furthermore, we have

Theorem 2.

if B is a proximal neighborhood of A in a proximity space (X,δ), then B is a (topological) neighborhood of A under the topologyMathworldPlanetmathPlanetmath τ(δ) induced by the proximity relation δ. In other words, if AB, then AB and AcB, where and c denote the interior and closure operatorsPlanetmathPlanetmathPlanetmath.

Proof.

Since AδB, then xδB whenever xA, which is the contrapositive of the statement: xA whenever xδB, which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to BcA, or AB. Furthermore, if xB, then xB. But AδB b assumptionPlanetmathPlanetmath. This implies xδA, which means xAc. Therefore AcB. ∎

Remark. However, not every τ(δ)-neighborhood is a δ-neighborhood.

Title proximal neighborhood
Canonical name ProximalNeighborhood
Date of creation 2013-03-22 16:58:25
Last modified on 2013-03-22 16:58:25
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 54E05
Synonym proximity neighborhood
Synonym δ-neighborhood