Extensions of zero-sets and of real-valued functions

Abstract

A Tychonoff space X which satisfies the property that G(X) = C(X δ ) is called an RG-space, where G(X) is the minimal regular ring extension of C(X) inside F (X), the ring of all functions from X to R, and X δ is the topology on X generated by its G δ -sets. We correct an error that we found in the proof of [19, Theorem 3.4

Extension properties and the Niemytzki plane

“…Proof. (1) ) (2): We prove this implication by induction on (A). If (A) = 0, it is obviously true since A = ?.…”

“…Finally, we brie y review scattered sets in R. Let (1) A is a scattered set in R; (2) A f0g is uniformly discrete in NP; (3) A f0g is P-embedded in NP; (4) A f0g is CU-embedded in NP. Proof.…”

“…Since the union of a uniformly discrete family of uniformly discrete subsets is also uniformly discrete, A f0g is uniformly discrete in NP. (2) ) (3) ) (4): Obvious. (4) ) (1): Suppose that A is not scattered; then A includes a perfect subset B which is closed in A.…”

“…The following theorem shows that every CU-embedded subset of NP is P-embedded, which answers Problem 1.2 for the Niemytzki plane negatively. (1) The set A = fx 2 R : hx; 0i 2 Y n Y 0 g is a scattered set in R; (2) Y n Y 0 is uniformly discrete in NP; (3) Y is P-embedded in NP; (4) Y is CU-embedded in NP.…”

A space which contains no realcompact dense subspace

“…If X is Oz, then every dense subspace is z-embedded in X. Scepin showed that every product of metrizable spaces is Oz. Blair and Hager [1] showed that a z-embedded subspace Y of a realcompact space X is realcompact if and only if Y is Gs-closed in X (i.e. for any point x in X -Y there is a Gs-set 5 of X such that x G S and S ft Y -0).…”