Topology

[Bart29]

For each of the following pairs A, B of topological spaces determine whether they are homeomorphic. In each case, either display a momeomorphism or give a convincing reason why they cannot be homeomorphic.

Note:
R=means real numbers
Q=stands for rational numbers
Z=stands for integers numbers
R\Q stands for is in R but not in Q.

a. A is the open interval (0,1) and B= R( both with the usual topology).
b. A is the open interval (0,1) and B= [0,1] (both with the usual topology)
c. A=R and B=R^2 (both with the usual topology).
( hint: what happens if you delete a point x_0 from R?)
d. A=Z and B=Q, both as subspaces of R.
e.A= (R\Q) and B=Q, both as subspaces of R.
f. A=R with the usual topology and B={(x,y) is a member of R^2: xy=1, x>0} as a subspace of R^2.

 

[Deleted member 4993]

For each of the following pairs A, B of topological spaces determine whether they are homeomorphic. In each case, either display a momeomorphism or give a convincing reason why they cannot be homeomorphic.

Note:
R=means real numbers
Q=stands for rational numbers
Z=stands for integers numbers
R\Q stands for is in R but not in Q.

a. A is the open interval (0,1) and B= R( both with the usual topology).
b. A is the open interval (0,1) and B= [0,1] (both with the usual topology)
c. A=R and B=R^2 (both with the usual topology).
( hint: what happens if you delete a point x_0 from R?)
d. A=Z and B=Q, both as subspaces of R.
e.A= (R\Q) and B=Q, both as subspaces of R.
f. A=R with the usual topology and B={(x,y) is a member of R^2: xy=1, x>0} as a subspace of R^2.

Are these part of a take of test?

Very well defined problem - where is your problem?

If you are stuck at the begining - please tell us the properties of the sets of rations
al, integers and real numbers.

Please share your work with us, indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[Bart29]

my friend tell me to help him with this problem he told me it was a hw problem. But couldnt help him. i havent taken topology. the problems are a-f questions.

 

[Deleted member 4993]

my friend tell me to help him with this problem he told me it was a hw problem. But couldnt help him. i havent taken topology. the problems are a-f questions.

To start off:

Please tell us the properties of metric spaces and Hausdorff spaces.

 

[Bart29]

Definition:A metric space(M,d) is a set M and a function d:MxM-->R such that

1.d(x,y) is greater than or equal to 0. for all x,y members of M
2. d(x,y)=0 iff x=y
3. d(x,y)=d(y,x) for every x,y members of M.
4. d(x,y) is less than or equal to d(x,z)+d(z,y) for all x,y, z members of M


Hausdorff
Definition:A topological space X satisfies the Hausdorff condition if for any two distinct points x,y members of X, there exist disjoint open sets U,V of X such that x is a member of U, y is a member of V.