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.

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Baire's Theorem Formulation

I am still having problems with some of the proofs of Baire's theorem. In Introductory Real Analysis by Kolmogorov and Fomin The statement of the theorem states that: "A complete metric space R ...
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Baire related problem

Let $f:\Bbb R\to \Bbb R$, $f$ in $C^{\infty}$. Suppose that for all $x \in \Bbb R$, there exists an integer $n$ (which depends on $x$) such that $f^{n}(x) = 0$ ($f^{n}$ is composing $f$ $n$ times) ...
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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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Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
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Equivalence of Forms of Baire Category Theorem

I am trying to show the equivalence of two forms of the Baire Category Theorem. These are the two statements: Let $(X,d)$ be a complete metric space. Let $U_n$ be a dense, open set for each $n \in ...
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Baire's Theorem proof regarding points revisited

My first question on this point was not answered. Here is the first part of Shilov's proof of Baire's theorem (not an exact lift from the book as I avoided mathematical symbols). I am trying to be ...
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Is this Baire theorem a special case of Baire category theorem?

I'm reading a chinese text book "Real Analysis" (by Zhou Minqiang), one of it's conclusion is "Baire theorem" For any $E\subset R^n$ is a $F_\sigma$ set: $$E=\bigcup_{k=1}^{\infty}F_k,$$ where ...
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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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For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
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Partition of $\mathbb{R}$ into nullset and 1st category set

Let $\{a_i\}_{i=1}^{\infty}$ be an enumeration of the rationals, $\mathbb{Q}$. Let $I_{ij}$ be the open interval centered at $a_i$ and having length $1/2^{i+j}$, and define $G_j = \cup_{i=1}^\infty ...
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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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Analogy for Baire categories?

I'm looking for an analogy to grasp the intuitive notion of size that Baire categories on $\mathbb{R}$ provides. For instance, the cardinality of a subset of $\mathbb{R}$ provides a notion of size in ...
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Baire Category Theorem with $G_{\delta}$

If we have a subset $Y$ of a complete metric space $X$ such that $Y$ is a $G_{\delta}$, then how would I go to show that $Y$ satisfies the Baire category theorem? I am trying to get there by showing ...
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Using Baire Category Theorem to prove $\mathbb{R}^2\not\cong\mathbb{R}^3$.

How can we prove $\mathbb R ^ 2$ is not homeomorphic to $\mathbb R ^3$ using Baire Category Theorem? Here is a standard proof of this fact using algebraic topology. Note that $\mathbb{R}^{3}-\{x\}$ ...
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Showing an operator kernel is nowhere dense…

We are doing a problem that requires us to, for fixed NONZERO element $h \in L^2[0,1]$ (as in, the function $h$ is not zero on a set of positive measure in $[0,1]$) and fixed $f \in L^2[0,1]$ with ...
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x-section of closure of E of first category implies x-section of E nowhere dense

Let $E$ be a subset of first category of product space $X \times Y$. Why is the following true: if $(\bar E)_x \subset Y$ is of first category then it follows that $E_x$ is nowhere dense. $E_x$ ...
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Continuity Points of The Deriviative of a Differentiable Function

I recently read the section on Baire Spaces from Munkres and this question came to my mind which I cannot settle. Does there exist a differentiable function $f:\mathbf R\to\mathbf R$ such that the ...
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A Property of Baire Spaces

Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is nowhere dense. It is easy to see that all Baire spaces have Property A. Is the ...
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Kuratowski-Ulam Theorem

I am looking for a proof of the Kuratowski-Ulam Theorem: Let $X,Y$ be Baire spaces, $Y$ be second countable and let $R \subseteq X \times Y$ have the Baire property. Then $\{ x \in X \ | \ R_x ...