Continuous Extensions of Functions Defined on Subsets of Products with the Κ-box Topology

Consider these results: (a) [N. Noble, Proc. Amer. Math.Soc. 31 (1972), 613–614] every Gδ-dense subspace in a product of separable metric spaces is C-embedded; (b) [Milton Don Ulmer, Ph.D. Dissertation. Wesleyan University (Middletown, CT, USA), 1970 and Pacific J. Math. 46 (1973), 591–602] every Σproduct in a product of first-countable spaces is C-embedded; (c) [R. Pol and E. Puzio-Pol, Fund. Math. 93 (1976), no. 1, 57–69; also A. V. Arhangel’skii, Topology Proc. 25 (2000), Summer, 383–416, 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 paper continues the first two authors’ earlier initiative [Topology Appl. 159 (2012), 2331–2337], which already generalized those cited results in several ways simultaneously, e.g., κ-box topology on the product spaces; relaxed separation properties on both the domain and the range spaces). Now the authors show: Let κ ≤ α satisfy λ < κ, β ≤ α ⇒ βλ ≤ α; let Y be dense in an open subset U of a κ-box product (Πi∈I Xi)κ with each Xi a T1-space; let q ∈ XI\Y have the property that for each J ∈ [I]≤α there is y ∈ Y such that yJ = qJ ; let Z be a regular space with a Gα+ -diagonal. Suppose that for each i ∈ I either χ(qi, Xi) ≤ α or each intersection of κ-many neighborhoods of qi is another such neighborhood. Then every continuous f : Y → Z extends continuously over Y ∪ {q}. Several corollaries and consequences are given. 2010 Mathematics Subject Classification. Primary 54B10, 54C45; Secondary 54G10.