Example of Set X with Two Topologies: Continuous But Not Homeomorphic

[kiriyama]

Give an example of a set X with two topologies T and S such that the identity function from (X,T) to (X,S) is continuous but not homeomorphic.

I always struggle with these because I get overwhelmed by the generality that it has. Any ideas would be very much appreciated.

 

[AKG]

What are the definitions of "continuous" and "homeomorphic"? Also, this post should be in the Homework Help section.

 

[kiriyama]

i apologize i am new here...just looked for the first place that seemed appropriate


and i assume the definitions to be the standard ones...

continuous:
T and S are topologies and F:X->Y; for each S open subset V of Y, f^-1(V) is a T open subset of X

and homeomorphic:
f is one to one, onto, continuous, and open

 

[AKG]

Okay, and if I is the identity function from (X,T) to (X,S), V is a subset of X, what is I^-1(V)? If U is a subset X, what is I(U)? Now if I is not a homeomorphism, it has to fail to have at least one of the four properties you listed under the definition of homeomorphic. If we are trying to find an example when I is continuous, then there is in fact only one of those four properties of homeomorphism that I would fail to have. Which is it?

 

[kiriyama]

i believe youre over complicating this

im looking for an example of a set X that has two topologies T and S that has an identity function from (X,T) to (X,S) that is continuous but not homeomorphic

i already know that given any space X and two topologies T and S, (X,T) and (X,S) are never homeomorphic.

im just looking for an example of this

 

[mathwonk]

you are looking for a set with two topologies, one contained in the other. ho hum.

try a 2 point set.

 

[kiriyama]

using a particular point topology?

or what?

can you please explain

i mean i get where youre headed but I am trying to find the missing step in between

 

[kiriyama]

nvmd...i got it...took a little longer than i hoped but i got it now

thanks for the help