On dimensionally restricted maps
Some results of B. Pasynkov and H. Torunczyk on finite-dimensional maps are improved. A generalization of a Dranishnikov-Uspenskij theorem about extensional dimension is also obtained.
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arXiv subjects
Vesko Valov
Publications and source records attributed to Vesko Valov.
Extension dimension and refinable maps
Extension dimension is characterized in terms of $ω$-maps. We apply this result to prove that extension dimension is preserved by refinable maps between metrizable spaces. It is also shown that refinable maps preserve some infinite-dimensional properties.
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Universal metric spaces and extension dimension
For any countable $CW$-complex $K$ and a cardinal number $τ\geqω$ we construct a completely metrizable space $X(K,τ)$ of weight $τ$ with the following properties: $\e X(K,τ)\leq K$, $X(K,τ)$ is an absolute extensor for all normal spaces $Y$ with $\e Y\leq K$, and for any completely metrizable space $Z$ of weight $\leqτ$ and $\e Z\leq K$ the set of closed embeddings $Z\to X(K,τ)$ is dense in the space $C(Z,X(K,τ))$ of all continuous maps from $Z$ into $X(K,τ)$ endowed with the limitation topology. This result is applied to prove the existence of universal spaces for all metrizable spaces of given weight and with a given cohomological dimension.
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