General Topology/Miscellaneous spaces
Noetherian spaces
editDefinition (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.
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 spaces
editDefinition (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:
- is irreducible
- Whenever are two closed subsets of that are both not all of , then
- Whenever is open and nonempty, it is dense
- 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 spaces
editExercises
edit- Suppose that is equipped with the cofinite topology. Prove that this topological space is irreducible, but does not admit a generic point.
- 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.