Turn on thread page Beta
    • Thread Starter
    Offline

    2
    ReputationRep:
    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?
    • PS Helper
    Offline

    14
    PS Helper
    Open subsets of T_1 \times \{ b \} are sets of the form U \times \{ b \} for U open in T_1, so if f : x \mapsto (x,b) then what is f^{-1}(U \times \{b \} )?

    P.S. Just to justify my classification of open sets, \{ b \} is a one-point space and so must have the discrete topology, i.e. the only open sets are \varnothing and \{ b \}, and clearly U \times \varnothing doesn't make much sense as an open set of T_1 \times \{b \} since (x,) doesn't make sense as an element.
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by nuodai)
    Open subsets of T_1 \times \{ b \} are sets of the form U \times \{ b \} for U open in T_1, so if f : x \mapsto (x,b) then what is f^{-1}(U \times \{b \} )?

    P.S. Just to justify my classification of open sets, \{ b \} is a one-point space and so must have the discrete topology, i.e. the only open sets are \varnothing and \{ b \}, and clearly U \times \varnothing doesn't make much sense as an open set of T_1 \times \{b \} since (x,) 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)?
    • PS Helper
    Offline

    14
    PS Helper
    (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 T_1 \times \{b \} as a subspace of T_1 \times T_2 will be the sets of the form (U \times V) \cap (T_1 \times \{ b \}) for U open in T_1 and V open in T_2, but this is equal to (U \cap T_1) \times (V \cap \{ b \}). Since U \subseteq T_1 and V \cap \{b \} = \{b\} (assuming b \in V), then this in turn is equal to U \times \{b \}. (If b \not \in V then we get \varnothing which is clearly open).
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: April 2, 2011
The home of Results and Clearing

995

people online now

1,567,000

students helped last year

University open days

  1. Keele University
    General Open Day Undergraduate
    Sun, 19 Aug '18
  2. University of Melbourne
    Open Day Undergraduate
    Sun, 19 Aug '18
  3. Sheffield Hallam University
    City Campus Undergraduate
    Tue, 21 Aug '18
Poll
A-level students - how do you feel about your results?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.