1. ## quasiorders

a quasiorder being defined as reflexive and transitive - not antisymmetric (if also antisymmetric then it is a partial ordering).

it is a trivial quasiorder iff: (Ax,y)(x<=y). can someone provide me with a few canonical, if not pathological, examples of a trivial quasiorder?

what is the relation between a neighborhood and an open set? a specialisation quasiorder .=. (x<=y) iff every neighborhood of x is a neighborhood of y. i need a few examples of a specialisation quasiorder.

why is a singleton necessarily a closed set -- why can't it be declared open?

thanks.

2. Originally Posted by nweissma
a quasiorder being defined as reflexive and transitive - not antisymmetric (if also antisymmetric then it is a partial ordering).

it is a trivial quasiorder iff: (Ax,y)(x<=y). can someone provide me with a few canonical, if not pathological, examples of a trivial quasiorder?

what is the relation between a neighborhood and an open set? a specialisation quasiorder .=. (x<=y) iff every neighborhood of x is a neighborhood of y. i need a few examples of a specialisation quasiorder.

why is a singleton necessarily a closed set -- why can't it be declared open?

thanks.
It's hard to think of a non-trivial example of trivial quasiorder. Define $\equiv$ by $\forall x,y, x \equiv y$. It's trivially trivial, but what you gonna do?

A neighborhood of a point x is any open set containing x. I've never heard of a specialization quasiorder.

There are valid topologies where a singleton is an open set. One example is the discrete topology where every set is an open set. It's a theorem that a topology satisfies the $T_1$ separation axiom iff every set containing a single point is closed.

3. ## neighborhoods vs open sets

A neighborhood of a point x is any open set containing x.

this is a mere clinical definition.

consider: separations axioms are defined in terms of open sets. can they be defined ito neighborhoods? how?