Path coverings with prescribed ends in faulty hypercubes

We discuss the existence of vertex disjoint path coverings with prescribed ends for the $n$-dimensional hypercube with or without deleted vertices. Depending on the type of the set of deleted vertices and desired properties of the path coverings we establish the minimal integer $m$ such that for every $n \ge m$ such path coverings exist. Using some of these results, for $k \le 4$, we prove Locke's conjecture that a hypercube with $k$ deleted vertices of each parity is Hamiltonian if $n \ge k +2.$ Some of our lemmas substantially generalize known results of I. Havel and T. Dvo\v{r}\'{a}k. At the end of the paper we formulate some conjectures supported by our results.Comment: 26 page

On the continuity of factorizations

[EN] Let {Xi : i ∈ I} be a set of sets, XJ :=Пi∈J Xi when Ø ≠ J ⊆ I; Y be a subset of XI , Z be a set, and f : Y → Z. Then f is said to depend on J if p, q ∈ Y , pJ = qJ ⇒ f(p) = f(q); in this case, fJ : πJ [Y ] → Z is well-defined by the rule f = fJ ◦ πJ|Y When the Xi and Z are spaces and f : Y → Z is continuous with Y dense in XI , several natural questions arise: (a) does f depend on some small J ⊆ I? (b) if it does, when is fJ continuous? (c) if fJ is continuous, when does it extend to continuous fJ : XJ → Z? (d) if fJ so extends, when does f extend to continuous f : XI → Z? (e) if f depends on some J ⊆ I and f extends to continuous f : XI → Z, when does f also depend on J? The authors offer answers (some complete, some partial) to some of these questions, together with relevant counterexamples. Theorem 1. f has a continuous extension f : XI → Z that depends on J if and only if fJ is continuous and has a continuous extension fJ : XJ → Z. Example 1. For ω ≤ k ≤ c there are a dense subset Y of [0, 1]k and f ∈ C(Y, [0, 1]) such that f depends on every nonempty J ⊆ k, there is no J ∈ [k]<ω such that fJ is continuous, and f extends continuously over [0, 1]k. Example 2. There are a Tychonoff space XI, dense Y ⊆ XI, f ∈ C(Y ), and J ∈ [I]<ω such that f depends on J, πJ [Y ] is C-embedded in XJ , and f does not extend continuously over XI .Comfort, W.; Gotchev, IS.; Recoder-Nuñez, L. (2008). On the continuity of factorizations. Applied General Topology. 9(2):263-280. doi:10.4995/agt.2008.1806.SWORD2632809

Continuous extensions of functions defined on subsets of products

AbstractA subset Y of a space X is Gδ-dense if it intersects every nonempty Gδ-set. The Gδ-closure of Y in X is the largest subspace of X in which Y is Gδ-dense.The space X has a regular Gδ-diagonal if the diagonal of X is the intersection of countably many regular-closed subsets of X×X.Consider now these results: (a) (N. Noble, 1972 [18]) every Gδ-dense subspace in a product of separable metric spaces is C-embedded; (b) (M. Ulmer, 1970 [22], 1973 [23]) every Σ-product in a product of first-countable spaces is C-embedded; (c) (R. Pol and E. Pol, 1976 [20], also A.V. Arhangelʼskiĭ, 2000 [3]; as corollaries of more general theorems), every dense subset of a product of completely regular, first-countable spaces is C-embedded in its Gδ-closure.The present authorsʼ Theorem 3.10 concerns the continuous extension of functions defined on subsets of product spaces with the κ-box topology. Here is the case κ=ω of Theorem 3.10, which simultaneously generalizes the above-mentioned results. TheoremLet {Xi:i∈I} be a set of T1-spaces, and let Y be dense in an open subspace of XI:=∏i∈IXi. If χ(qi,Xi)⩽ω for every i∈I and every q in the Gδ-closure of Y in XI, then for every regular space Z with a regular Gδ-diagonal, every continuous function f:Y→Z extends continuously over the Gδ-closure of Y in XI.Some examples are cited to show that the hypothesis χ(qi,Xi)⩽ω cannot be replaced by the weaker hypothesis ψ(qi,Xi)⩽ω