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


  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.