You are Here: Home >< Maths

# Question about topological spaces watch

1. http://www2.imperial.ac.uk/~tsorense...lemSheet10.pdf

Question 5(i). Let's prove that T1 is homemorphic to T1 x {b} (the other one will be similar). I constructed the obvious bijection mapping x in T1 to (x,b) in T1 x {b}. But how can I prove that it is continuous? I tried to do it by definition (pre-image of an open set) but I just got bogged down by the definitions - since this has the subspace topology induced from the product space with the product topology! Is there any easy way to deal with this?
2. Open subsets of are sets of the form for open in , so if then what is ?

P.S. Just to justify my classification of open sets, is a one-point space and so must have the discrete topology, i.e. the only open sets are and , and clearly doesn't make much sense as an open set of since doesn't make sense as an element.
3. (Original post by nuodai)
Open subsets of are sets of the form for open in , so if then what is ?

P.S. Just to justify my classification of open sets, is a one-point space and so must have the discrete topology, i.e. the only open sets are and , and clearly doesn't make much sense as an open set of since doesn't make sense as an element.
You put the product topology on T1 x {b}. Is that definitely the same as thinking of T1 x {b} as a subspace of T1 x T2 and giving it the subspace topology (this is what confused me, because you have subspaces and products together in the same question)?
4. (Original post by gangsta316)
You put the product topology on T1 x {b}. Is that definitely the same as thinking of T1 x {b} as a subspace of T1 x T2 and giving it the subspace topology (this is what confused me, because you have subspaces and products together in the same question)?
Well you'll get the same thing either way. A basis for the subspace topology on as a subspace of will be the sets of the form for open in and open in , but this is equal to . Since and (assuming ), then this in turn is equal to . (If then we get which is clearly open).

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: April 2, 2011
The home of Results and Clearing

### 995

people online now

### 1,567,000

students helped last year
Today on TSR

### University open days

1. Keele University
Sun, 19 Aug '18
2. University of Melbourne
Sun, 19 Aug '18
3. Sheffield Hallam University
Tue, 21 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams