Let X and Y be metric compacta such that there exists a continuous open surjection from $C_p(Y)$ onto $C_p(X)$. We prove that if there exists an integer k such that $X^k$ is strongly infinite-dimensional, then there exists an integer p such that $Y^p$ is strongly infinite-dimensional.
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M. Steinberger et J. West ont prouvé dans [7] qu'un fibré de Serre p:E → B entre CW-complexes a la propriété de relèvement des homotopies par rapport aux k-espaces. Malheureusement, leur démonstration contient une légère erreur. Ils affirment que certains ensembles (notés U et $p^{-1}U×U$) sont des CW-complexes car ce sont des ouverts de CW-complexes. Ceci est généralement faux, et notre premier objectif dans cette note est de donner des exemples d'ouverts de CW-complexes n'admettant aucune décomposition CW. Malgré cela, le théorème de Steinberger et West est vrai, et notre deuxième objectif est de montrer comment leur démonstration peut être rectifiée.
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We prove that a k-dimensional hereditarily indecomposable metrisable continuum is not a $P_k$-valued absolute retract. We deduce from this that none of the classical characterizations of ANR (metric) extends to the class of stratifiable spaces.
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For A ⊂ I = [0,1], let $L_A$ be the set of continuous real-valued functions on I which vanish on a neighborhood of A. We prove that if A is an analytic subset which is not an $F_σ$ and whose closure has an empty interior, then $L_A$ is homeomorphic to the space of differentiable functions from I into ℝ.
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Let D (resp. D*) be the subspace of C = C([0,1], R) consisting of differentiable functions (resp. of functions differentiable at the one point at least). We give topological characterizations of the pairs (C, D) and (C, D*) and use them to give some examples of spaces homeomorphic to C\D or to C\D*.
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We give an example in the Hilbert space $ℓ^2$ of two subsets which are absorbing for the class of topologically complete spaces, but for which there exists no homeomorphism of $ℓ^2$ onto itself mapping one of these subsets onto the other.
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Given a pair (M,X) of spaces we investigate the connections between the (strong) universality of (M,X) and that of the space X. We apply this to prove Enlarging, Deleting, and Strong Negligibility Theorems for strongly universal and absorbing spaces. Given an absorbing space Ω we also study the question of topological uniqueness of the pair (M,X), where $M = [0,1]^{ω}$ or $M = (0,1)^{ω}$ and X is a copy of Ω in M having a locally homotopy negligible complement in M.
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We prove that a space M with Disjoint Disk Property is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. This implies that the product M × I² of a space M with the disk is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. The proof of these theorems exploits the homological characterization of Q-manifolds due to Daverman and Walsh, combined with the existence of G-stable points in C-spaces. To establish the existence of such points we prove (and afterward apply) homological versions of the Brouwer Fixed Point Theorem and of Uspenskij's Selection Theorem.
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Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power $X^{ω}$ of any subspace X ⊂ Y is not universal for the class 𝓐₂ of absolute $G_{δσ}$-sets; moreover, if Y is an absolute $F_{σδ}$-set, then $X^{ω}$ contains no closed topological copy of the Nagata space 𝓝 = W(I,ℙ); if Y is an absolute $G_{δ}$-set, then $X^{ω}$ contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power $X^{ω}$ of any absolute retract of the first Baire category contains a closed topological copy of each σ-compact space having a strongly countable-dimensional completion. We also prove that for a Polish space X and a subspace Y ⊂ X admitting an embedding into a σ-compact sigma hereditarily disconnected space Z the weak product $W(X,Y) = {(x_i) ∈ X^{ω}: almost all x_i ∈ Y} ⊂ X^{ω}$ is not universal for the class ℳ ₃ of absolute $G_{δσδ}$-sets; moreover, if the space Z is compact then W(X,Y) is not universal for the class ℳ ₂ of absolute $F_{σδ}$-sets.
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