General Topology/Miscellaneous spaces

Noetherian spacesEdit

Definition (noetherian topological space):

Let   be a topological space.   is called noetherian if and only if for all ascending chains of open subsets

 

there exists   such that for all  , we have  .

In the latter case, we say that the ascending chain   stabilizes.

Proposition (noetherian iff all open sets are compact):

Let   be a topological space.   is noetherian if and only if all of its open subsets are compact.

(On the condition of the axiom of dependent choice.)

Proof: Suppose first that   is noetherian, and let   be open. Let   be an open cover of  . By definition of the subspace topology, each   is open in  . An open cover of   is constructed as thus: Pick   arbitrary. Once   are chosen, either we already have  , or we may select   such that  . This process must terminate, or else, upon defining

 ,

we obtain an ascending chain

 

which does not stabilize. Suppose now that all open subsets of   are compact. Let

 

be any ascending chain of open subsets of  , and define

 .

We immediately see that   is an open cover of  , so that by its compactness, we may extract a finite subcover   for certain indices  . Now set   so that for  

 , ie.  ,

that is, the ascending chain stabilizes.  

Irreducible spacesEdit

Definition (irreducible space):

Let   be a topological space.   is called irreducible or hyperconnected if whenever   are open and nonempty, then  .

Proposition (characterisation of irreducible spaces):

Let   be a topological space. Then the following are equivalent:

  1.   is irreducible
  2. Whenever   are two closed subsets of   that are both not all of  , then  
  3. Whenever   is open and nonempty, it is dense
  4. Whenever   is closed, it is nowhere dense

Proof: We prove 1.   2.   3.   4.   1. Let first   be irreducible, and suppose that   are two proper closed subsets of  . Suppose that  , and define   and  . Then  . Suppose now that   is open and nonempty and 2. holds. If   is open, but not dense,   is not all of  , and further   is closed and not all of   (  was nonempty). Therefore,  , the union of two proper closed subsets, which is impossible by 2. Suppose now 3. holds and   is closed. Then   is open and hence dense in  . Let   be an arbitrary open subset, and suppose that   is dense in  .  

Definition (generic point):

Let   be a topological space. A generic point is an element   such that  .

Proposition (a generic point is contained in every open subset):

Let   be a topological space, and let   be a generic point of  . Whenever   is open,  .

That is, every open set contains every generic point.

Proof: Suppose  . Then   is a superset of the closure of  , in contradiction to  .  

Definition (sober):

A topological space   is called sober iff every closed and irreducible subset of   admits a unique generic point.

Local spacesEdit

ExercisesEdit

  1. Suppose that   is equipped with the cofinite topology. Prove that this topological space is irreducible, but does not admit a generic point.
  2. Prove that on the two-point space   one may find a topology that makes   into an irreducible space with two generic points. Generalize this example to any set.