Where are typical $C^{1}$ functions one-to-one?

Zoltán Buczolich; András Máthé

Displaying similar documents to “Where are typical $C^{1}$ functions one-to-one?”

Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of ℝⁿ

Anders Nilsson, Peter Wingren (2007)

Studia Mathematica

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A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have $(d i m_{H} (U), {\underset{̲}{d i m}}_{B} (U), {\bar{d i m}}_{B} (U)) = (r, s, t)$ . Moreover, $2^{- n} \leq H^{r} (K) \leq 2 n^{r / 2}$ .

A poset of topologies on the set of real numbers

Vitalij A. Chatyrko, Yasunao Hattori (2013)

Commentationes Mathematicae Universitatis Carolinae

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On the set $ℝ$ of real numbers we consider a poset $𝒫_{τ} (ℝ)$ (by inclusion) of topologies $τ (A)$ , where $A \subseteq ℝ$ , such that $A_{1} \supseteq A_{2}$ iff $τ (A_{1}) \subseteq τ (A_{2})$ . The poset has the minimal element $τ (ℝ)$ , the Euclidean topology, and the maximal element $τ (\emptyset)$ , the Sorgenfrey topology. We are interested when two topologies $τ_{1}$ and $τ_{2}$ (especially, for $τ_{2} = τ (\emptyset)$ ) from the poset define homeomorphic spaces $(ℝ, τ_{1})$ and $(ℝ, τ_{2})$ . In particular, we prove that for a closed subset $A$ of $ℝ$ the space $(ℝ, τ (A))$ is homeomorphic to the Sorgenfrey line $(ℝ, τ (\emptyset))$ iff $A$ is countable. We study also common...

Köthe dual of Banach sequence spaces $ℓ_{p} [X]$ $(1 \leq p < \infty)$ and Grothendieck space

Cong Xin Wu, Qing Ying Bu (1993)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we show the representation of Köthe dual of Banach sequence spaces $ℓ_{p} [X]$ $(1 \leq p < \infty)$ and give a characterization of that the spaces $ℓ_{p} [X]$ $(1 < p < \infty)$ are Grothendieck spaces.

Compacta are maximally $G_{δ}$ -resolvable

István Juhász, Zoltán Szentmiklóssy (2013)

Commentationes Mathematicae Universitatis Carolinae

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It is well-known that compacta (i.e. compact Hausdorff spaces) are maximally resolvable, that is every compactum $X$ contains $Δ (X)$ many pairwise disjoint dense subsets, where $Δ (X)$ denotes the minimum size of a non-empty open set in $X$ . The aim of this note is to prove the following analogous result: Every compactum $X$ contains $Δ_{δ} (X)$ many pairwise disjoint $G_{δ}$ -dense subsets, where $Δ_{δ} (X)$ denotes the minimum size of a non-empty $G_{δ}$ set in $X$ .

Indiscernibles and dimensional compactness

C. Ward Henson, Pavol Zlatoš (1996)

Commentationes Mathematicae Universitatis Carolinae

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This is a contribution to the theory of topological vector spaces within the framework of the alternative set theory. Using indiscernibles we will show that every infinite set $u S \subseteq G$ in a biequivalence vector space $〈 W, M, G 〉$ , such that $x - y \notin M$ for distinct $x, y \in u$ , contains an infinite independent subset. Consequently, a class $X \subseteq G$ is dimensionally compact iff the $π$ -equivalence $≐_{M}$ is compact on $X$ . This solves a problem from the paper [NPZ 1992] by J. Náter, P. Pulmann and the second author.

Displaying similar documents to “Where are typical C 1 functions one-to-one?”

Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of ℝⁿ

A poset of topologies on the set of real numbers

Köthe dual of Banach sequence spaces ℓ p [ X ] ( 1 ≤ p < ∞ ) and Grothendieck space

Compacta are maximally G δ -resolvable

Indiscernibles and dimensional compactness

Displaying similar documents to “Where are typical $C^{1}$ functions one-to-one?”

Köthe dual of Banach sequence spaces $ℓ_{p} [X]$ $(1 \leq p < \infty)$ and Grothendieck space

Compacta are maximally $G_{δ}$ -resolvable