Questions tagged [baire-category]

This tag is intended for questions on topics related to Baire category, such as Baire category theorem, meager sets (set of first category), nonmeager sets (set of second category), Baire spaces etc.

The Baire category theorem (BCT) asserts that a countable intersection of open dense sets is still dense. A topological space for which the BCT holds is called a Baire space. Examples include locally compact spaces and complete metric spaces. The BCT has major applications in real and functional analysis, such as the open mapping theorem, or the existence of many nowhere differentiable functions.

A meager set is a set which is a countable union nowhere dense sets. They are also called sets of first category. Non-meager sets are called sets of second category. An equivalent formulation of the BCT is that all open subsets of a topological space $X$ are sets of second category.

The theorem is provable from the Zermelo–Fraenkel set theory with the axiom of choice, $\sf ZFC$, but not from $\sf ZF$ (in fact BCT is equivalent to the Principle of Dependent Choice, or $\sf DC$, over $\sf ZF$).

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A subset of the second Baire category on the real line

Why is the subset in $\mathbb{R}$ of the second Baire category uncountable?
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Problem seeing how Baire's theorem applies in a proof of open mapping theorem

I cant see which version and how they use Baires theorem to get that atleast on $MB_{n}$ is dense in some open set. Any version of Baires theorem needs open or closed sets. I can get neither on the $MB_{n}$'s.
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Every comeager set in a perfect Polish space contains an uncountable dense $G_\delta$ set

Let $X$ be a perfect Polish Space. Prove that every comeager contains an uncountable dense $G_\delta$ set. It's known that every perfect Polish Space has cardinality $2^{\aleph_0}$. It's easy to see that every comeager contains a $G_\delta$ set,…
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