Let $X = [0,1]^{\omega_1}$ (that is, all functions from the set of countable ordinals to the unit interval) have the product topology. Let $Y$ be all functions in $X$ with countable support (that is, all points $(x_\alpha) \in X$ with $x_\alpha = 0$ for all but countably many $\alpha \in \omega_1$). I am trying to find an example to illustrate a somewhat technical point and, if I am correct in thinking that $Y$ as above is nonmeagre, then my job will be done. Unfortunately, I'm not sure how to approach this. Certainly I don't see any obvious way to express $Y$ as a countable union of nowhere dense subsets of $X$...
Am I correct in thinking this set is nonmeagre?
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general-topology
1 Answers
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This $Y$ is the so-called $\Sigma$-product of $\omega_1$ many copies of $[0,1]$ w.r.t. $0$. It is countably compact and dense in the full product $X$, and so by lemma 2.2.17 of this syllabus (e.g.) $Y$ intersects every non-empty $G_\delta$ subset of $X$, and this implies that, as $X$ is a Baire space, so is $Y$ (it's exercise 2.2.15 in the same syllabus). So this confirms your suspicion. $Y$ is an interesting space: it's normal countably compact, non-compact, ccc but not separable.
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0@mike: It's called a $\Sigma$-product in a lot of papers, and e.g. in Engelking's general topology, though it is only discussed in exercises, IIRC. Quite standard in general topology. If instead of countably many non-zero (in general, different coordinates than the base point) we allow only finitely many, we get a so-called $\sigma$-product. They are also naturally topological groups whenever we take a product of topological groups and use the "all identity" vector as basepoint, and they also occur in functional analysis (Corson compact subsets etc.) – 2011-01-27