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.
What are the definitions of "continuous" and "homeomorphic"? Also, this post should be in the Homework Help section.
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
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?
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
you are looking for a set with two topologies, one contained in the other. ho hum. try a 2 point set.
using a particular point topology? or what? can you please explain i mean i get where youre headed but im trying to find the missing step in between