- #1

MathematicalPhysicist

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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?