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If $\mu^\star|P(X)$ would be a measure, then we could define a $\sigma$-additive measure $\lambda:P([0,1])\to[0,1]$ by the formula $\lambda(A)=\mu^\star(A\cap X)$ for $A\subset [0,1]$. This would impl …

This question has a negative answer (given by Gregorz Plebanek), which follows from the following theorem of Gitik and Shelah. Theorem (Gitik-Shelah, 1989): If a set $X$ admits an atomless probabilit …

The answer to this problem is Yes. Indeed, the inclusion $E\subset \bigcap_{C\in\mathcal C}E_C$ is trivial, so it remains to prove that for any point $x\in\mathbb R^n\setminus E$ there exists a conne …

In fact, this problem has been answered affirmatively in this paper of T.Banakh, M.Filipczak and J.Wodka, and also by MO-user Dap in his comment to this MO question.

A counterexample to this problem can be constructed as follows. Take a sequence $(K_n)_{n\in\omega}$ of pairwise disjoint nowhere dense compact sets $K_n\subset[0,1]$ of positive Lebesgue measure $\l …

This question is equivalent to the question of Willie Wong because of the following theorem of Jones. Theorem (Jones, 1967). Each bijective semicontinuous map from a topological space to a semilo …

After some thinking I realized that the answer to this question is negative. A counterexample can be constructed by a standard diagonal method of killing all possible candidatures. We shall construct …

Let $B$ be a compact convex set on the complex plane, containing zero in its interior. The boundary $\partial B$ of $B$ has the polar parametrization $\mathbf p:\mathbb R\to \partial B$ assigning to e …

Problem. What is the Borel complexity of the set $$c(\mathbb Q)=\{(x_n)_{n\in\omega}\in\mathbb R^\omega:\exists\lim_{n\to\infty}x_n\in\mathbb Q\}$$ in the countable product of lines $\mathbb R^\omega$ …

This is in Exercise 23.11 in the textbook of Kechris (and follows from the $\mathbf \Pi^0_3$-completeness of this space). The topological (infinite-dimensional) structure of this space is described in …

The answer is No. A suitable counterexample can be constructed as follows. On the real line $\mathbb R$ consider the equivalence relation $E=\{(x,y)\in\mathbb R\times \mathbb R:x-y\in\mathbb Q\}$. …

The sequentiality does not match well with an algebraic structure. For example, the following result of Banakh and Zdomskyy characterizes sequential topological groups with countable $cs^*$-character: …

Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with …

Question. Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ whose restriction $f|\mathbb Q^\omega$ is injective?

Let us call a series $\sum_n x_n$ in a Banach space "good" if there exists a permutation $\sigma:\mathbb N\to\mathbb N$ such that the rearranged series $\sum_n x_{\sigma(n)}$ converges. Find a simpl …