Examples for seperation axioms.

[MathematicalPhysicist]

My assignment is like this:
1.give an example of a space X and a subspace A of X s.t X satisifes Sep and A doesnt.
2.give an example of a continuous and onto function f:X->Y s.t X satisifies S1 but Y doesnt.
3.give an example of a continuous and onto function f:X->Y s.t X satisfies S2 and Y doesnt.

my answers are as follows:
1. X=R A=R-Q, is it a good example?
2.X=N, Y=N-{0}U{sqrt2} and f:X->Y f(x)=x if x in N-{0} and f(x)=sqrt2 if x=0, i think we can't find a countable set of bases for Y, not sure though.
3. didn't do it so far, any hints?

 

[HallsofIvy]

First, what do you mean by "separation"? I would consider all of the "Tychonoff" properties to be "separation" properties: T0: each singleton set is closed; T1: given any two points, there exist a set that contains one of them but not the other; T2 (Hausdorf): Given any two points there exist two disjoint open set such that one contains one point and the other set contains the other point.

 

[MathematicalPhysicist]

Sep says there's a countable dense set.
S1 says there's a countable basis at a point.
S2 says there's a countable basis for the topology.

those should be the countability axioms, sorry for misleading.

 

[morphism]

"1." isn't good. The irrationals are actually a separable subspace of R (in the usual topology). In fact, if X is a separable metric space, then any subspace of X is separable as well. So your example is going to have to come from a non-metrizable topology.

For 2, what topology are you giving Y?

For 3, I would think about using X=Y, but giving it two topologies, one finer than the other. Then maybe using f=identity.