induced partial order on an Alexandroff space

Let X be a T0, Alexandroff space. For AX denote by Ao the intersectionMathworldPlanetmath of all open neighbourhoods of A. Define a relationMathworldPlanetmath on X as follows: for any x,yX we have xy if and only if x{y}o. This relation will be called the induced partial orderMathworldPlanetmath on X.

PropositionPlanetmathPlanetmath 1. (X,) is a poset.

Proof. Of course x{x}o for any xX. Thus is reflexiveMathworldPlanetmathPlanetmath.

Assume now that xy and yx for some x,yX. Assume that xy. Then, since X is a T0 space, there is an open set U such that xU and yU or there is an open set V such that yV and xV. Both cases lead to contradictionMathworldPlanetmathPlanetmath, because we assumed that x{y}o and y{x}o. Thus every open neighbourhood of one element must also contain the other. Thus is antisymmetric.

Finally assume that xy and yz for some x,y,zX. Since y{z}o, then {z}o is an open neighbourhood of y and thus {y}o{z}o. Therefore x{z}o, so is transitiveMathworldPlanetmathPlanetmathPlanetmath, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Proposition 2. Let X,Y be two, T0, Alexandroff spaces and f:XY be a function. Then f is continuousPlanetmathPlanetmath if and only if f preserves the induced partial order.

Proof. ,,” Assume that f is continuous and suppose that x,yX are such that xy. We wish to show that f(x)f(y), so assume this is not the case. Let A={f(y)}o. Then f(x)A. But A is open, so f-1(A) is also open (because we assumed that f is continuous). Furthermore yf-1(A) and because xy, then xf-1(A), but this implies that f(x)A. Contradiction.

,,” Assume that f preserves the induced partial order and let UY be an open subset. Let xU. Then for any yx we have f(y)f(x) (because f preserves the induced partial order) and since {f(x)}oU (because U is open and {f(x)}o is the smallest open neighbourhood of f(x)) we have that f(y)U. Thus


which implies that f-1(U) is open because f-1(A) contains a small neighbourhood of each point. This completes the proof.

Title induced partial order on an Alexandroff space
Canonical name InducedPartialOrderOnAnAlexandroffSpace
Date of creation 2013-03-22 18:45:55
Last modified on 2013-03-22 18:45:55
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Derivation
Classification msc 54A05