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Math Help - Box and Product Topologies

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    Super Member Aryth's Avatar
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    Box and Product Topologies

    I actually have two questions. One about topological comparisons between those two topologies and one about the identity map. First, I just wanted to make sure a proof I was using was correct or not... I've battled with it for awhile and I want some criticism.

    The exercise was: Let \{(\mathbb{R}_i,\Omega_i) : i \in \mathbb{N}\} be an indexed family of copies of the real line under the usual topology and let \mathbb{R}^{\infty} = \prod\{\mathbb{R}_i : i \in \mathbb{N}\}. Show that the product topology is a proper subset of the box topology on \mathbb{R}^{\infty}.

    My proof was this:

    For this proof, it will suffice to show that P \subset B where B = \prod\{ U_i : U_i \in \Omega_i\} and P = \prod \{ V_i : V_i \in \Omega_i \text{ and all but finitely many } V_i \neq \mathbb{R}_i\} are the bases of the Box and Product topologies, respectively. We will first consider U_i \subset \mathbb{R}_i. We know, then, that \prod\{U_i : i\in \mathbb{N}\} \in B but not in P since every set in P has U_i = \mathbb{R}_i for all but a finite number of U_i. Thus, B \not\subseteq P. Now we will show that P \subset B. We first remember that, in P, any, all or no V_i from \{V_i : i\in \mathbb{N}\} will be \mathbb{R}_i. B only requires that each individual U_i \in \Omega_i, and this is true whether U_i \subseteq \mathbb{R}_i or U_i = \mathbb{R}_i. Hence, we can conclude that any element of P is in B and P \subset B.

    The second question was this:

    For each i \in \mathbb{Z}^+, let f_i : \mathbb{R} \to \mathbb{R} be the identity map f_i(x) = x. Show that each f_i is continuous relative to the usual topology on \mathbb{R}.

    I am not sure how to proceed with this. Any hints or help would be much appreciated.
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    Super Member Aryth's Avatar
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    Re: Box and Product Topologies

    I realize now that the second question was obvious... So, strike that one. I have to present this proof to my class, so it would be great if I could get some criticism on it.
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