# Tagged Questions

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Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$. Is $S\times S$ homeomorphic to $S$? By Luzin ...
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### Showing a filter with a certain property on the power set of $\mathbb{Z}$ is a one point filter

Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let  \mathcal{A} := \{ f \in ...
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### When is the class of functions between sets a set?

I'm reading the paper 'The big fundamental group, big Hawaiian earrings and the big free groups'. The authors state that the class of homotopy equivalences of loops in the space he dubs as the big ...
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### Axiom of Choice and Continuous function

Do you know if the folowing statement is an equivalent form of AC or not ?? *If $X$ is a compact metric space then every continuous function $f: X \longrightarrow \mathbb{R}$ is uniformly ...
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### Injective with Finite Discontinuities Mapping from $\mathbb R^n$ to $[0,1]$

Hi, as a continuation to the fully answered question: Injective&Intregrable Mapping from $\mathbb R^3$ to $\mathbb R$ Can one think of an injective $f:\mathbb R^n\rightarrow[0,1]$ that has only ...
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### Injective&Intregrable Mapping from $\mathbb R^3$ to $\mathbb R$

Hi, Can one think of an injective and Riemann integrable $f:\mathbb R^3\rightarrow\mathbb R$? (of course it cannot be continuous)
Upd. The question in a nutshell: find convex set on plane which is «closest» to a given non-convex set. It is given integrable function $0\leq f(x,y)\leq 1$ with bounded support: $f(x,y)=0$ when ...