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I'm doing this exercise in Munkres book and got no clue about the solution. Hope some one can help me solve this.

Show that the product space $R^{I}$, where $I = [0,1]$, has a countable dense subset. If $J$ has cardinality greater than $\mathscr{P}(Z_{+})$, then the product space $R^{J}$ does not have a countable dense subset

With this problem, I even can't imagine the space $R^{I}$. So if some one can talk something about this space, I really appreciate it (as I know, this space is rather important, right?)

Thanks so much

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  • $\begingroup$ Actually, $\mathbb{R}^I$ is the space of all functions $I\to \mathbb{R}$ with the topology of pointwise convergence. I don't know if this can help you, though. $\endgroup$ – Giuseppe Negro Sep 9 '13 at 17:29
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    $\begingroup$ You might find the answers to this question useful: math.stackexchange.com/questions/420384/… $\endgroup$ – BaronVT Sep 9 '13 at 17:50
  • $\begingroup$ There is Hewitt—Marczewski—Pondiczery theorem which says $Π_{i < 2^κ} X_i$ has density $≤ κ$ if all $X_i$ have desity $≤ κ$. $\mathbb{R}^I$ is product of continuum separable spaces hence is separable. On the other hand the limit of $2^κ$ spaces in HMP theorem is maximal so product of more than continuum separable spaces is needs not to be separable. $\endgroup$ – user87690 Sep 9 '13 at 21:26
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Note that the structure of $I$ or $J$ is irrelevant for topology of $\mathbb{R}^I$ or $\mathbb{R}^J$. Only the cardinality matters. So we want to show that product of continuum copies of $\mathbb{R}$ is separable but product of more than continuum copies of $\mathbb{R}$ is not separable. Actually the following holds: Product of $≤ 2^κ$ spaces of density $≤ κ$ has density $≤ κ$ (for $κ$ infinite cardinal). This is called Hewitt–Marczewski—Pondiczery theorem (e. g. Engelking 2.3.15). On the other hand if all spaces forming a product contain two disjoint open subsets (e. g. at least 2 point Hausdorff spaces) then there is injection from index set to powerset of a dense set so product of more than $2^κ$ such spaces cannot have density $≤ κ$. For proofs and more details see http://dantopology.wordpress.com/2009/11/06/product-of-separable-spaces/.

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