# Tagged Questions

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### A homeomorphism of a topological space with itself maps a set into one of the same category

Prove: If $h$ is a homeomorphism of $S$ onto $S$ and if $E\subset S$, then $E$ and $h(E)$ have the same category in $S$. Rudin, Functional Analysis, 2/e, p.43. (My own answer follows below.)
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### Is complement of a 1st category set in a 2nd category space, dense?

Let $X$ be a second category space and $Y$ be a first category subspace of $X$. Is $X \setminus Y$ dense in $X$?
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### B meager, not empty, not open implies complement is dense

Let X be topological space. The subset B is meager, not open and non-empty. How can you prove that the complement is dense?
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### how do i prove that the the set of irrationals cannot be a countable union of closed subsets? [duplicate]

Let $\mathbb{R}$ be equipped with the standard topology. Let $E$ be the set of irrational numbers. How do i prove that $E$ is not a countable union of closed subsets, using Baire Category Theorem?
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### Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
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### If every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space

Show that if every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space. (Munkres "Topology", 48.3) Here is what I tried : Let $\{U_n\}_{n \geq 1}$ be a collection of ...
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### corollary to baire category theorem

I'm studying topology with gamelin and greene's text and I came across a corollary to the baire category theorem which states that "Let (En) be a sequence of nowhere dense subsets of a complete ...
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### Equivalence of Baire Space definitions

I am hoping someone could help me show that the following statements, which define a Baire Space, are equivalent. Defn1: Any topological space X such that the intersection of any countable ...
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### Baire space but not locally compact

I need an example that is a Baire space but not locally compact. I think, $\mathbb{R}^ \mathbb{R}$ is such an example. $\mathbb{R}^ \mathbb{R}$ is not locally compact. But I could not proof that it ...
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### In a Baire space $X$, if an open set meets a nonmeager set, is the intersection nonmeager?

In a Baire space $X$, if an open set $U$ meets a nonmeager set $N$, is the intersection nonmeager? If they do not meet then it's false; take the upper half line of the reals. It does not meet some of ...
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### $X$ a separable metric space with no isolated points. If $G \subset X$ is a countable dense $G_{\delta}$ subset of $X$, why is $X$ meager?

Let $X$ be a separable metric space with no isolated points. Then if $G \subset X$ is a countable dense $G_{\delta}$ subset of $X$, why is $X$ meager? I've written a couple things so far, Let ...
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### Examples of closed subspaces of Baire spaces that fail to be Baire?

I am looking for some nice examples of Baire spaces containing closed subspaces that fail to be Baire. Clearly, $X$ should not satisfy either one of the standard hypotheses for the Baire category ...
### $X$ is a complete metric space, $Y$ is compact. $X \times Y$ is Baire?
Requesting a hint or solution. X is a complete metric space and Y is a compact hausdorff space. Trying to show that $X \times Y$ is a Baire space.