neighborhood system on a set


In point-set topology, a neighborhood system is defined as the set of neighborhoodsMathworldPlanetmathPlanetmath of some point in the topological spaceMathworldPlanetmath.

However, one can start out with the definition of a β€œabstract neighborhood system” 𝔑 on an arbitrary set X and define a topologyMathworldPlanetmath T on X based on this system 𝔑 so that 𝔑 is the neighborhood system of T. This is done as follows:

Let X be a set and 𝔑 be a subset of XΓ—P⁒(X), where P⁒(X) is the power setMathworldPlanetmath of X. Then 𝔑 is said to be a abstract neighborhood system of X if the following conditions are satisfied:

  1. 1.

    if (x,U)βˆˆπ”‘, then x∈U,

  2. 2.

    for every x∈X, there is a UβŠ†X such that (x,U)βˆˆπ”‘,

  3. 3.

    if (x,U)βˆˆπ”‘ and UβŠ†VβŠ†X, then (x,V)βˆˆπ”‘,

  4. 4.

    if (x,U),(x,V)βˆˆπ”‘, then (x,U∩V)βˆˆπ”‘,

  5. 5.

    if (x,U)βˆˆπ”‘, then there is a VβŠ†X such that

    • –

      (x,V)βˆˆπ”‘, and

    • –

      (y,U)βˆˆπ”‘ for all y∈V.

In additionPlanetmathPlanetmath, given this 𝔑, define the abstract neighborhood system around x∈X to be the subset 𝔑x of 𝔑 consisting of all those elements whose first coordinate is x. Evidently, 𝔑 is the disjoint unionMathworldPlanetmath of 𝔑x for all x∈X. Finally, let

T = {UβŠ†X∣for everyΒ x∈U,Β (x,U)βˆˆπ”‘}
= {UβŠ†X∣for everyΒ x∈U, there is aΒ VβŠ†U, such thatΒ (x,V)βˆˆπ”‘}.

The two definitions are the same by condition 3. We assert that T defined above is a topology on X. Furthermore, Tx:={U∣(x,U)βˆˆπ”‘x} is the set of neighborhoods of x under T.

Proof.

We first show that T is a topology. For every x∈X, some UβŠ†X, we have (x,U)βˆˆπ”‘ by condition 2. Hence (x,X)βˆˆπ”‘ by condition 3. So X∈T. Also, βˆ…βˆˆT is vacuously satisfied, for no xβˆˆβˆ…. If U,V∈T, then U∩V∈T by condition 4. Let {Ui} be a subset of T whose elements are indexed by I (i∈I). Let U=⋃Ui. Pick any x∈U, then x∈Ui for some i∈I. Since Ui∈T, (x,Ui)βˆˆπ”‘. Since UiβŠ†U, (x,U)βˆˆπ”‘ by condition 3, so U∈T.

Next, suppose 𝒩 is the set of neighborhoods of x under T. We need to show 𝒩=Tx:

  1. 1.

    (π’©βŠ†Tx). If Nβˆˆπ’©, then there is U∈T with x∈UβŠ†N. But (x,U)βˆˆπ”‘, so by condition 3, (x,N)βˆˆπ”‘, or (x,N)βˆˆπ”‘x, or N∈Tx.

  2. 2.

    (TxβŠ†π’©). Pick any U∈Tx and set W={z∣U∈Tz}. Then x∈WβŠ†U by condition 1. We show W is open. This means we need to find, for each z∈W, a VβŠ†W such that (z,V)βˆˆπ”‘. If z∈W, then (z,U)βˆˆπ”‘. By condition 5, there is Vβˆˆπ”‘ such that (z,V)βˆˆπ”‘, and for any y∈V, (y,U)βˆˆπ”‘, or y∈U by condition 1. So y∈W by the definition of W, or VβŠ†W. Thus W is open and Uβˆˆπ’©.

This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof. By the way, W defined above is none other than the interior of U: W=U∘. ∎

Remark. Conversely, if T is a topology on X, we can define 𝔑x to be the set consisting of (x,U) such that U is a neighborhood of x. The the union 𝔑 of 𝔑x for each x∈X satisfies conditions 1 through 5 above:

  1. 1.

    (condition 1): clear

  2. 2.

    (condition 2): because (x,X)βˆˆπ”‘ for each x∈X

  3. 3.

    (condition 3): if U is a neighborhood of x and V a supserset of U, then V is also a neighborhood of x

  4. 4.

    (condition 4): if U and V are neighborhoods of x, there are open A,B with x∈AβŠ†U and x∈BβŠ†V, so x∈A∩BβŠ†U∩V, which means U∩V is a neighborhood of x

  5. 5.

    (condition 5): if U is a neighborhood of x, there is open A with x∈AβŠ†U; clearly A is a neighborhood of x and any y∈A has U as neighborhood.

So the definition of a neighborhood system on an arbitrary set gives an alternative way of defining a topology on the set. There is a one-to-one correspondence between the set of topologies on a set and the set of abstract neighborhood systems on the set.

Title neighborhood system on a set
Canonical name NeighborhoodSystemOnASet
Date of creation 2013-03-22 16:41:34
Last modified on 2013-03-22 16:41:34
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 54-00
Defines abstract neighborhood system