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Homework Help: [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!
     
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