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A connection between multiplication in C(X) and the dimension of X

Andrzej Komisarski (2006)

Fundamenta Mathematicae

Let X be a compact Hausdorff topological space. We show that multiplication in the algebra C(X) is open iff dim X < 1. On the other hand, the existence of non-empty open sets U,V ⊂ C(X) satisfying Int(U· V) = ∅ is equivalent to dim X > 1. The preimage of every set of the first category in C(X) under the multiplication map is of the first category in C(X) × C(X) iff dim X ≤ 1.

A dimension raising hereditary shape equivalence

Jan Dijkstra (1996)

Fundamenta Mathematicae

We construct a hereditary shape equivalence that raises transfinite inductive dimension from ω to ω+1. This shows that ind and Ind do not admit a geometric characterisation in the spirit of Alexandroff's Essential Mapping Theorem, answering a question asked by R. Pol.

A dimensional property of Cartesian product

Michael Levin (2013)

Fundamenta Mathematicae

We show that the Cartesian product of three hereditarily infinite-dimensional compact metric spaces is never hereditarily infinite-dimensional. It is quite surprising that the proof of this fact (and this is the only proof known to the author) essentially relies on algebraic topology.

A factorization theorem for the transfinite kernel dimension of metrizable spaces

M. Charalambous (1998)

Fundamenta Mathematicae

We prove a factorization theorem for transfinite kernel dimension in the class of metrizable spaces. Our result in conjunction with Pasynkov's technique implies the existence of a universal element in the class of metrizable spaces of given weight and transfinite kernel dimension, a result known from the work of Luxemburg and Olszewski.

A Non-Probabilistic Proof of the Assouad Embedding Theorem with Bounds on the Dimension

Guy David, Marie Snipes (2013)

Analysis and Geometry in Metric Spaces

We give a non-probabilistic proof of a theorem of Naor and Neiman that asserts that if (E, d) is a doubling metric space, there is an integer N > 0, depending only on the metric doubling constant, such that for each exponent α ∈ (1/2; 1), one can find a bilipschitz mapping F = (E; dα ) ⃗ ℝ RN.

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