Seminarium 01.10.2026
Taras Banakh
Zapraszamy wszystkich chętnych do przedstawienia otwartych (lub rozwiązanych) problemów ich pracy naukowej oraz do wspólnej dyskusji.
Seminarium ?.10.2026
Magdalena Nowak
For a metric Peano continuum $X$ of a diameter $\le 1$, let $S_X$ be the function assigning to each $\varepsilon>0$ the smallest cardinality of a cover of $X$ by connected subsets of diameter $\le \varepsilon$.
We show that for any strictly increasing function $\Omega:\mathbb{R}_+\to\mathbb{R}_+$ with $(0,1]\subseteq\Omega[\mathbb{R}_+]$ and $$s:=\sum_{n=1}^\infty S_X(\tfrac{1}{2^{n}})\sum_{m=n}^\infty S_X({\tfrac{1}{2^{m}}})\cdot\Omega^{-1}(\min\{1,\tfrac{1}{2^{m-6}}\})<\infty,$$ there exists a continuous surjective function $f\colon[0,s]\to X$ with continuity modulus $\omega_f\le\Omega$.
This controlled version of the Hahn-Mazurkiewicz Theorem implies that $$SDim(X)\le HDim(X)\le 2{\cdot} SDim(X),$$ where $SDim(X)=\limsup_{\varepsilon\to 0}\frac{\ln(S_X(\varepsilon))}{\ln(1/\varepsilon)}$ is the Sierpiński dimension of $X$, and $$HDim(X)=\inf\{\alpha\in (0,\infty]:\;\mbox{there is a~surjective $\frac1\alpha$-Hölder map $f:[0,1]\to X\}$}$$ is the Hölder dimension of $X$.
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Seminarium ...01.2027
