1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

[Topology] Product Spaces

  1. Jan 19, 2010 #1
    [Topology] Product Spaces :(

    1. The problem statement, all variables and given/known data

    1. Show that in the product space [tex]N^N[/tex] where the topology on N is discrete, the set of near-constant functions is dense (near constant function is a function that becomes constant from a specific index..)...

    2. Prove that in [tex]R^I[/tex] the set of monotonic increasing functions is not open.

    2. Relevant equations
    3. The attempt at a solution

    I've no idea how to start thinking of these questions....

    I'll be delighted to receive some guidance

    Thanks in advance
     
  2. jcsd
  3. Jan 19, 2010 #2
    Re: [Topology] Product Spaces :(

    1) A set is dense if its closure is the whole space. That is, if for every point [itex]f \in \mathbb{N}^\mathbb{N}[/itex] every neighborhood of f intersects the set, but it's sufficient to consider all subbasis elements containing f since all other open neighborhoods can be formed with intersections and unions of these. Let D denote the set of near-constant functions and consider an element [itex]f \in \mathbb{N}^\mathbb{N}[/itex]. Consider a subbasis element U that contains f. Its nth component is [itex]\mathbb{N}[/itex] except for at one index k. Now define,
    [tex]g(x) = \begin{cases} f(x) & \textrm{if }x = k \\ 0 & \textrm{otherwise} \end{cases}[/tex]
    Then, g is in D and in the neighborhood U of f.

    2) If the set of monotonic increasing functions were open then, then every monotonically increasing function would have an open neighborhood that consists only of monotonic increasing functions. However if U is such a neighborhood of a monotonically increasing function f, then U is [itex]\mathbb{R}[/itex] at all points except for finitely many. Thus you can choose a function g in U that is not monotonically increasing by choosing a sufficiently small value at one of the points where U is [itex]\mathbb{R}[/itex]. For instance if U is [itex]\mathbb{R}[/itex] at [itex]m > 0[/itex], then consider,
    [tex]g(x) = \begin{cases} f(0)-1 & \textrm{if }x=m \\ f(x) & \textrm{otherwise} \end{cases}[/tex]
     
  4. Jan 20, 2010 #3
    Re: [Topology] Product Spaces :(

    Thanks a lot! You're very helpful!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook