Let $E$ be an infinite set of positive integers, proves that there is a $\alpha \in \mathbb{R}$ such that $\{\left \lfloor \alpha^k \right \rfloor ;k \in \mathbb{N} \}\cap E$ is infinite. I have two ideas in mind, first I can try to use $\lfloor \alpha^k \rfloor=n \iff \log(\alpha)\in [\frac{log(n)}{k},\frac{log(n+1)}{k})$ in order to prove that some […]

This is problem 5 in section 27 of Munkres’ TOPOLOGY, 2nd ed Let $X$ be a compact Hausdorff space; let $\{A_n\}_{n\in \mathbb{N}}$ be a countable collection of closed sets of $X$. If each set $A_n$ has empty interior in $X$, then the union $\bigcup A_n$ has empty interior in $X$. How to show this fact? […]

Let $f$ belong to $ C^{\infty}[0,1]$ and for each $x \in [0,1]$ there exists $n \in \mathbb{N}$ so that $f^{(n)}(x)=0$. Prove that $f$ is a polynomial in $[0,1]$. I am trying to use Baire Category Theorem , but cannot perfectly complete the proof . Thanks for any help.

We say that a function $f$ is Baire Class $1$ if there is a sequence of functions $f_i \to f$ pointwise where each $f_i$ is continuous. The set of discontinuities of a Baire Class $1$ function $f$ must be a first-category (meagre) set of points. The Dirichlet function $f(x) = \begin{cases} 1 & x \in […]

There is this theorem called the Banach category theorem which states that in every topological space any union of open sets of first category is of first category. Now doesn’t this imply that every topological space X (let’s say which is not of first category) is the union of a set of first category (the […]

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 copy: Intersections of open sets and $\alpha$-favorable spaces I was curious a little about the classes of topological spaces described below. For each of […]

I have seen this interesting paragraph on a talk page of the Wikipedia article about Polish mathematician Tomek Bartoszyński: Tomek’s paper “Additivity of measure implies additivity of category, Trans. Amer. Math. Soc., 281(1984), no. 1, 209–213″ as the first to show that there is (in ZFC) an asymmetry between measure and category. Before that essentially […]

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 classical consequences of the Baire category theorem in basic functional analysis — the open mapping theorem, the closed graph theorem and the uniform […]

I’m asking a question in reference to Noah Schweber’s answer in this question can open set in $\mathbb{R}^n$ be written as the union of countable disjoint closed sets?, which is itself a reference to GEdgar’s answer here: Does not exist cover of $\mathbb{R}^n$ by disjoint closed balls. I’m having trouble with the second condition: showing […]

A number $x \in (0,1]$ written in non-terminating dyadic expansion $$ \sum_{n\ge 1} \frac{e_n(x)}{2^n} $$ is said to be normal if $\frac{1}{n}(e_1(x)+\cdots+e_n(x))\to \frac{1}{2}$ as $n\to \infty$. Let $N$ be the set of normal numbers. It is known that $N$ is a set of measure one and, at the same time, a meager set (see the […]

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