() @ Appl. Gen. Topol. 16, no. 1(2015), 65-74doi:10.4995/agt.2015.3161 c© AGT, UPV, 2015 On C-embedded subspaces of the Sorgenfrey plane Olena Karlova Chernivtsi National University, Department of Mathematical Analysis, Kotsjubyns’koho 2, Cher- nivtsi 58012, Ukraine (maslenizza.ua@gmail.com) Abstract We prove that every C∗-embedded subset of S2 is a hereditarily Baire subspace of R2. We also show that for a subspace E ⊆ {(x, −x) : x ∈ R} of the Sorgenfrey plane S2 the following conditions are equivalent: (i) E is C-embedded in S2; (ii) E is C∗-embedded in S2; (iii) E is a countable Gδ-subspace of R 2 and (iv) E is a countable functionally closed subspace of S2. 2010 MSC: 54C45; 54C20. Keywords: C ∗-embedded; C-embedded; the Sorgenfrey plane. 1. Introduction Recall that a subset A of a topological space X is called functionally open (functionally closed) in X if there exists a continuous function f : X → [0, 1] such that A = f−1((0, 1]) (A = f−1(0)). Sets A and B are completely separated in X if there exists a continuous function f : X → [0, 1] such that A ⊆ f−1(0) and B ⊆ f−1(1). A subspace E of a topological space X is • C-embedded (C∗-embedded) in X if every (bounded) continuous function f : E → R can be continuously extended on X; • z-embedded in X if every functionally closed set in E is the restriction of a functionally closed set in X to E; • well-embedded in X [7] if E is completely separated from any function- ally closed set of X disjoint from E. Received 3 July 2014 – Accepted 5 December 2014 http://dx.doi.org/10.4995/agt.2015.3161 O. Karlova Clearly, every C-embedded subspace of X is C∗-embedded in X. The con- verse in not true. Indeed, if E = N and X = βN, then E is C∗-embedded in X (see [4, 3.6.3]), but the function f : E → R, f(x) = x for every x ∈ E, does not extend to a continuous function f : X → R. A space X has the property (C∗ = C) [11] if every closed C∗-embedded subset of X is C-embedded in X. The classical Tietze-Urysohn Extension Theorem says that if X is a normal space, then every closed subset of X is C∗- embedded and X has the property (C∗ = C). Moreover, a space X is normal if and only if every its closed subset is z-embedded (see [9, Proposition 3.7]). The following theorem was proved by Blair and Hager in [2, Corollary 3.6]. Theorem 1.1. A subset E of a topological space X is C-embedded in X if and only if E is z-embedded and well-embedded in X. A space X is said to be δ-normally separated [10] if every closed subset of X is well-embedded in X. The class of δ-normally separated spaces includes all normal spaces and all countably compact spaces. Theorem 1.1 implies the following result. Corollary 1.2. Every δ-normally separated space has the property (C∗ = C). According to [15] every C∗-embedded subspace of a completely regular first countable space is closed. The following problem is still open: Problem 1.3 ([12]). Does there exist a first countable completely regular space without property (C∗ = C)? H. Ohta in [11] proved that the Niemytzki plane has the property (C∗ = C) and asked does the Sorgenfrey plane S2 (i.e., the square of the Sorgenfrey line S) have the property (C∗ = C)? In the given paper we obtain some necessary conditions on a set E ⊆ S2 to be C∗-embedded. We prove that every C∗-embedded subset of S2 is a hereditarily Baire subspace of R2. We also characterize C- and C∗-embedded subspaces of the anti-diagonal D = {(x, −x) : x ∈ R} of S2. Namely, we prove that for a subspace E ⊆ D of S2 the following conditions are equivalent: (i) E is C- embedded in S2; (ii) E is C∗-embedded in S2; (iii) E is a countable Gδ-subspace of R2 and (iv) E is a countable functionally closed subspace of S2. 2. Every finite power of the Sorgenfrey line is a hereditarily α-favorable space Recall the definition of the Choquet game on a topological space X between two players α and β. Player β goes first and chooses a nonempty open subset U0 of X. Player α chooses a nonempty open subset V1 of X such that V1 ⊆ U0. Following this player β must select another nonempty open subset U1 ⊆ V1 of X and α must select a nonempty open subset V2 ⊆ U1. Acting in this way, the players α and β obtain sequences of nonempty open sets (Un) ∞ n=0 and (Vn) ∞ n=1 such that Un−1 ⊆ Vn ⊆ Un for every n ∈ N. The player α wins if ∞ ⋂ n=1 Vn 6= ∅. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 66 On C-embedded subspaces of the Sorgenfrey plane Otherwise, the player β wins. If there exists a rule (a strategy) such that α wins if he plays according to this rule, then X is called α-favorable. Respectively, X is called β-unfavorable if the player β has no winning strategy. Clearly, every α-favorable space X is β-unfavorable. Moreover, it is known [13] that a topological space X is Baire if and only if it is β-unfavorable in the Choquet game. If A is a subspace of a topological space X, then A and intA mean the closure and the interior of A in X, respectively. Lemma 2.1. Let X = n ⋃ k=1 Xk, where Xk is an α-favorable subspace of X for every k = 1, . . . , n. Then X is an α-favorable space. Proof. We prove the lemma for n = 2. Let G = G1 ∪ G2, where Gi = intXi, i = 1, 2. We notice that for every i = 1, 2 the space Xi is α-favorable, since it contains dense α-favorable subspace. Then Gi is α-favorable as an open subspace of the α-favorable space Xi. It is easy to see that the union G of two open α-favorable subspaces is an α-favorable space. Therefore, X is α- favorable, since G is dense in X. � Let p = (x, y) ∈ R2 and ε > 0. We write B[p; ε) = [x, x + ε) × [y, y + ε), B(p; ε) = (x − ε, x + ε) × (y − ε, y + ε). If A ⊆ S2 then the symbol clS2A (clR2A) means the closure of A in the space S2 (R2). We say that a space X is hereditarily α-favorable if every its closed subspace is α-favorable. Theorem 2.2. For every n ∈ N the space Sn is hereditarily α-favorable. Proof. Let n = 1 and ∅ 6= F ⊆ S. Assume that β chose a nonempty open in F set U0 = [a0, b0)∩F , a0 ∈ F . If U0 has an isolated point x in S, then α chooses V1 = {x} and wins. Otherwise, α put V1 = [a0, c0) ∩ F , where c0 ∈ (a0, b0) ∩ F and c0 − a0 < 1. Now let U1 = [a1, b1) ∩ F ⊆ V1 be the second turn of β such that a1 ∈ F and the set (a1, b1) ∩ F has no isolated points in S. Then there exists c1 ∈ (a1, b1) ∩ F such that c1 − a1 < 1 2 . Let V2 = [a1, c1) ∩ F . Repeating this process, we obtain sequences (Um) ∞ m=0, (Vm) ∞ m=1 of open subsets of F and sequences of points (am) ∞ m=0, (bm) ∞ m=0 and (cm) ∞ m=1 such that [am, bm) ⊇ [am, cm) ⊇ [am+1, bm+1), cm − am < 1 m+1 , cm ∈ F , Um = [am, bm) ∩ F and Vm+1 = [am, cm) ∩ F for every m = 0, 1, . . . . According to the Nested Interval Theorem, the sequence (cm) ∞ m=1 is convergent in S to a point x ∗ ∈ ∞ ⋂ m=0 Vm. Since F is closed in S, x∗ ∈ F . Hence, F ∩ ∞ ⋂ m=0 Vm 6= ∅. Consequently, F is α-favorable. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 67 O. Karlova Suppose that the theorem is true for all 1 ≤ k ≤ n and prove it for k = n+1. Consider a set ∅ 6= F ⊆ Sn+1. Let the player β chooses a set U0 = F ∩ n+1 ∏ k=1 [a0,k, b0,k) with a0 = (a0,k) n+1 k=1 ∈ F . Denote U+0 = n+1 ∏ k=1 (a0,k, b0,k) and consider the case U+0 ∩ F = ∅. For every k = 1, . . . , n + 1 we set U0,k = {a0,k}× ∏ i6=k [a0,i, b0,i) and F0,k = F ∩U0,k. Since U0,k is homeomorphic to S n, by the inductive assumption the space F0,k is α-favorable for every k = 1, . . . , n+1. Then F is α-favorable according to Lemma 2.1. Now let U+0 ∩ F 6= ∅. If there exists an isolated in Sn+1 point x ∈ U0, then α put V1 = {x} and wins. Assume U0 has no isolated points in S n+1. Then there is c0 = (c0,k) n+1 k=1 ∈ U+0 ∩ F such that diam( n+1 ∏ k=1 [a0,k, c0,k)) < 1. We put V1 = F ∩ n+1 ∏ k=1 [a0,k, c0,k). Let U1 = F ∩ n+1 ∏ k=1 [a1,k, b1,k) be the second turn of β such that a1 = (a1,k) n+1 k=1 ∈ F and U1 ⊆ V1. Again, if U + 1 ∩ F = ∅, where U + 1 = n+1 ∏ k=1 (a1,k, b1,k), then, using the inductive assumption, we obtain that for every k = 1, . . . , n + 1 the space F ∩ ( {a1,k} × ∏ i6=k [a1,i, b1,i) ) is α-favorable. Then α has a winning strategy in F by Lemma 2.1. If U+1 ∩F 6= ∅ and U1 has no isolated points in S n+1, the player α chooses a point c1 = (c1,k) n+1 k=1 ∈ U+1 ∩F such that diam( n+1 ∏ k=1 [a1,k, c1,k)) < 1/2 and put V2 = F ∩ n+1 ∏ k=1 [a1,k, c1,k). Repeating this process, we obtain sequences of points (am) ∞ m=0, (bm) ∞ m=0 and (cm) ∞ m=0, and of sets (Um) ∞ m=0 and (Vm) ∞ m=1, which satisfy the following properties: 1) Um = F ∩ n+1 ∏ k=1 [am,k, bm,k); 2) am ∈ F , cm ∈ U + m ∩ F ; 3) Vm+1 = F ∩ n+1 ∏ k=1 [am,k, cm,k); 4) Vm+1 ⊆ Um ⊆ Vm; 5) diam(Vm+1) < 1 m+1 for every m = 0, 1, . . . . We observe that the sequence (cm) ∞ m=0 is convergent in Rn+1 and x∗ = lim m→∞ cm ∈ ∞ ⋂ m=0 Vm = ∞ ⋂ m=0 Vm. Since cm → x ∗ in Sn+1, cm ∈ F and F is closed in Sn+1, x∗ ∈ F ∩ ( ∞ ⋂ m=0 Vm ) . Hence, F is α-favorable. � c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 68 On C-embedded subspaces of the Sorgenfrey plane 3. Every C∗-embedded subspace of S2 is a hereditarily Baire subspace of R2. Lemma 3.1. A set E ⊆ R2 is functionally closed in S2 if and only if 1) E is Gδ in R 2; and 2) if F is R2-closed set disjoint from E, then F and E are completely separated in S2. Proof. Necessity. Let f : S2 → R be a continuous function such that E = f−1(0). According to [1, Theorem 2.1], f is a Baire-one function on R2. Con- sequently, E is a Gδ subset of R 2. Condition (2) follows from the fact that every R2-closed set is, evidently, a functionally closed subset of S2. Sufficiency. Since E is Gδ in R 2, there exists a sequence of R2-closed sets Fn such that X \ E = ∞ ⋃ n=1 Fn. Clearly, E ∩Fn = ∅. Then condition (2) implies that for every n ∈ N there exists a continuous function fn : S 2 → R such that E ⊆ f−1n (0) i Fn ⊆ f −1(1). Then E = ∞ ⋂ n=1 f−1n (0). Hence, E is functionally closed in S2. � Lemma 3.2. Let X be a metrizable space, A ⊆ X be a set without isolated points and let B ⊆ X be a countable set such that A ∩ B = ∅. Then there exists a set C ⊆ A without isolated points such that C ∩ B = ∅. Proof. Let d be a metric on X, which generates its topological structure. For x0 ∈ X and r > 0 we denote B(x0, r) = {x ∈ X : d(x, x0) < r} and B[x0, r] = {x ∈ X : d(x, x0) ≤ r}. Let B = {bn : n ∈ N}. We put A0 = ∅ and construct sequences (An) ∞ n=1 and (Vn) ∞ n=1 of nonempty finite sets An ⊆ A and open neighborhoods Vn of bn which for every n ∈ N satisfy the following conditions: An−1 ⊆ An;(3.1) ∀x ∈ An ∃y ∈ An \ {x} with d(x, y) ≤ 1 n ;(3.2) d(An, ⋃ 1≤i≤n Vi) > 0.(3.3) Let A1 = {x1, y1}, where d(x1, y1) ≤ 1 and x1 6= y1. We take ε > 0 such that A1 ∩ B[b1, ε] = ∅ and put V1 = B(b1, ε). Assume that we have al- ready defined finite sets A1, . . . Ak and neighborhoods V1, . . . , Vk of b1, . . . , bk, respectively, which satisfy conditions (3.1)–(3.3) for every n = 1, . . . , k. Let Ak = {a1, . . . , am}, m ∈ N. Taking into account that the set D = A \ ⋃ 1≤i≤k V i has no isolated points, for every i = 1, . . . , m we take ci ∈ D with ci 6= ai and d(ai, ci) ≤ 1 k+1 . Put Ak+1 = Ak ∪ {c1, . . . , cm}. Take δ > 0 such that Ak+1 ∩ B[bk+1, δ] = ∅. Let Vk+1 = B(bk+1, δ). Repeating this process, we obtain needed sequences (An) ∞ n=1 and (Vn) ∞ n=1. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 69 O. Karlova It remains to put C = ∞ ⋃ n=1 An. � The following results will be useful. Theorem 3.3 ([5]). A subspace E of a topological space X is C∗-embedded in X if and only if every two disjoint functionally closed subsets of E are completely separated in X. Theorem 3.4 ([16]). The Sorgenfrey plane S2 is strongly zero-dimensional, i.e., for any completely separated sets A and B in S2 there exists a clopen set U ⊆ S2 such that A ⊆ U ⊆ S2 \ B. Recall that a space X is hereditarily Baire if every its closed subspace is Baire. Theorem 3.5. Let E be a C∗-embedded subspace of S2. Then E is a heredi- tarily Baire subspace of R2. Proof. Assume that E is not R2-hereditarily Baire space and take an R2-closed countable subspace E0 without R 2-isolated point (see [3]). Notice that E is S2-closed according to [15, Corollary 2.3]. Therefore, E0 is S 2-closed set. By Theorem 2.2 the space E0 is α-favorable, and, consequently, E0 is a Baire subspace of S2. Let E′0 be a set of all S 2-nonisolated points of E0. Since E ′ 0 is the set of the first category in S2-Baire space E0, the set G = E0\E ′ 0 is S 2-dense open discrete subspace of E0. We notice that G is R 2-dense subspace of E0. By Lemma 3.2 there exists a set C ⊆ G without R2-isolated point such that clR2C ∩ E ′ 0 = ∅. We put F = clR2C ∩ E0. Let A and B be any R2-dense in F disjoint sets such that F = A ∪ B. Evidently A and B are clopen subsets of F , since F is S2-discrete space. Notice that F is z-embedded in E, because F is countable. Moreover, F is R2-closed in E. Hence, F is S2-functionally closed in E. By Theorem 1.1 the set F is C-embedded in C∗-embedded in S2 set E. Consequently, F is C∗-embedded in S2. Therefore, Theorem 3.3 and Theorem 3.4 imply that there exist disjoint clopen set U, V ⊆ S2 such that A = U ∩ F and B = V ∩ F . According to Lemma 3.1 the sets U and V are Gδ in R 2. Let D = clR2F . Then U ∩ D and V ∩ D are R2-dense in D disjoint Gδ-sets, which contradicts to the baireness of D. � 4. Every discrete C∗-embedded subspace of S2 is a countable Gδ-subspace of R 2 . Lemma 4.1. Let X be a metrizable separable space and A ⊆ X be an uncount- able set. Then there exists a set Q ⊆ A which is homeomorphic to the set Q of all rational numbers. Proof. Let A0 be the set of all points of A which are not condensation points A (a point a ∈ X is called a condensation point of A in X if every neighborhood of a contains uncountably many elements of A). Notice that A0 is countable, c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 70 On C-embedded subspaces of the Sorgenfrey plane since X has a countable base. Put B = A \ A0. Then the inequality |A| > ℵ0 implies that every point of B is a condensation point of B. Take a countable subset Q ⊆ B which is dense in B. Clearly, every point of Q is not isolated. Hence, Q is homeomorphic to Q by the Sierpiński Theorem [14]. � Lemma 4.2. Let E be an R2-hereditarily Baire z-embedded subspace of S2. Then the set E0 of all isolated points of E is at most countable. Proof. Assume E0 is uncountable. Notice that E0 is an Fσ-subset of E, since E0 is an open subset of E and S2 is a perfect space by [6]. Then E0 = ∞ ⋃ n=1 En, where every set En is closed in E. Take N ∈ N such that EN is uncountable. According to Lemma 4.1 there exists a set Q ⊆ EN which is homeomorphic to Q. Since Q is clopen in EN and EN is a clopen subset of a z-embedded in S 2 set E, there exists a functionally closed subset Q1 of S 2 such that Q = E ∩Q1. By Lemma 3.1 the set Q1 is a Gδ-set in R 2. Then Q is a Gδ-subset of a hereditarily Baire space E. Hence, Q is a Baire space, a contradiction. � Theorem 4.3. If E is a discrete C∗-embedded subspace of S2, then E is a countable Gδ-subspace of R 2. Proof. Theorem 3.5 and Lemma 4.2 imply that E is a countable hereditarily Baire subspace of R2. According to [8, Proposition 12] the set E is Gδ in R2. � The converse implication in Theorem 4.3 is not valid as Theorem 4.5 shows. Lemma 4.4. Let A be an S2-closed set, ε > 0 and L(A; ε) = {p ∈ S2 : B[p; ε) ⊆ A}. Then L(A; ε) is R2-closed. Proof. We take p0 = (x0, y0) ∈ clR2L(A; ε) and show that p0 ∈ L(A; ε). We consider U = intR2B[p0; ε) and prove that U ⊆ A. Take p = (x, y) ∈ U and put δ = min{(x − x0)/2, (y − y0)/2, (x0 + ε − x)/2, (y0 + ε − y)/2}. Let p1 ∈ B(p0; δ) ∩ L(A; ε). It is easy to see that p ∈ B[p1; ε). Then p ∈ A, since p1 ∈ L(A; ε). Hence, U ⊆ A. Then B[p0; ε) = clS2U ⊆ clS2A = A, which implies that p0 ∈ L(A; ε). Therefore, L(A; ε) is closed in R 2. � Theorem 4.5. There exists an S2-closed countable discrete Gδ-subspace E of R2 which is not C∗-embedded in S2. Proof. Let C be the standard Cantor set on [0, 1] and let (In) ∞ n=1 be a sequence of all complementary intervals In = (an, bn) to C such that diam (In+1) ≤ diam (In) for every n ≥ 1. We put pn = (bn; 1 − an), E = {pn : n ∈ N} and F = {(x, 1 − x) : x ∈ R} ∩ (C × [0, 1]). Notice that E is a closed subset of S2, F is functionally closed in S2 and E ∩ F = ∅. Let N′ ⊆ N be a set such that {bn : n ∈ N ′} and {bn : n ∈ N\ N ′} are dense subsets of C. To show that E is not C∗-embedded in S2 we verify that disjoint clopen subsets E1 = {pn : n ∈ N ′} and E2 = {pn : n ∈ N \ N ′} c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 71 O. Karlova of E can not be separated by disjoint clopen subsets in S2. Assume the contrary and take disjoint clopen subsets W1 and W2 of S 2 such that Wi ∩ E = Ei for i = 1, 2. We prove that W1 ∩ F is R 2-dense in F . To obtain a contradiction we take an R2-open set O such that O ∩ F ∩ W1 = ∅. Since the set U = S 2 \ W1 is clopen, U = ∞ ⋃ n=1 L(U; 1 n ), where L(U; 1 n ) = {p ∈ S2 : B[p; 1/n) ⊆ U} and the set Fn = L(U; 1 n ) is R2-closed by Lemma 4.4 for every n ∈ N. Since O ∩ F is a Baire subspace of R2, there exist N ∈ N and an R2-open in F subset I ⊆ F such that I ∩ O ⊆ FN ∩ F ⊆ S 2 \ E1. Taking into account that diam (In) → 0, we choose n1 > N such that bn − an < 1 2N for all n ≥ n1. Since the set {an : n ∈ N ′} is dense in C, there exists n2 ∈ N ′ such that n2 > n1 and p = (an2; 1 − an2) ∈ I. Clearly, p ∈ F . Consequently, B[p; 1 N ) ∩ E1 = ∅. But pn2 ∈ B[p, 1 N ) ∩ E1, a contradiction. Similarly we can show that W2 ∩ F is also R 2-dense in F . Notice that W1 and W2 are Gδ in R 2 by Lemma 3.1. Hence, W1 ∩ F and W2 ∩ F are disjoint dense Gδ-subsets of a Baire space F , which implies a contradiction. Therefore, E is not C∗-embedded in S2. � 5. A characterization of C-embedded subsets of the anti-diagonal of S2. By D we denote the anti-diagonal {(x, −x) : x ∈ R} of the Sorgenfrey plane. Notice that D is a closed discrete subspace of S2. Theorem 5.1. For a set E ⊆ D the following conditions are equivalent: 1) E is C-embedded in S2; 2) E is C∗-embedded in S2; 3) E is a countable Gδ-subspace of R 2; 4) E is a countable functionally closed subspace of S2. Proof. The implication (1) ⇒ (2) is obvious. The implication (2) ⇒ (3) follows from Theorem 4.3. We prove (3) ⇒ (4). To do this we verify condition (2) from Lemma 3.1. Let F be an R2-closed set disjoint from E. Denote D = F ∩D and U = ⋃ p∈D B[p; 1). We show that U is clopen in S2. Clearly, U is open in S2. Take a point p0 ∈ clS2U and show that p0 ∈ U. Choose a sequence pn ∈ U such that pn → p0 in S 2. For every n there exists qn ∈ D such that pn ∈ B[qn, 1). Notice that the sequence (qn) ∞ n=1 is bounded in R 2 and take a convergent in R2 subsequence (qnk) ∞ k=1 of (qn) ∞ n=1. Since D is R 2-closed, q0 = lim k→∞ qnk ∈ D. Then p0 ∈ clR2B[q0, 1). If p0 ∈ B[q0, 1), then p0 ∈ U. Assume p0 6∈ B[q0, 1) and let q0 = (x0, y0). Without loss of generality we may suppose that p0 ∈ [x0, x0 + 1] × {y0 + 1}. Since pnk → p0 in S 2, qnk ∈ (−∞, x0] × [y0, +∞) for all k ≥ k0 and p0 ∈ [x0, x0 + 1) × {y0 + 1}. Then p0 ∈ ⋃∞ k=1 B[qnk , 1) ⊆ U. Hence, U is clopen and D = U ∩ D. Since D and F \ U are disjoint functionally closed c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 72 On C-embedded subspaces of the Sorgenfrey plane subsets of S2, there exists a clopen set V such that D ∩ V = ∅ and F \ U ⊆ V . Then F ⊆ U ∪ V ⊆ S2 \ E. Consequently, F and E are completely separated in S2. Therefore, E is functionally closed in S2 by Lemma 3.1. (4) ⇒ (1). Notice that E satisfy the conditions of Theorem 1.1. Indeed, E is z-embedded in S2, since |E| ≤ ℵ0. Moreover, E is well-embedded in S 2, since E is functionally closed. � Remark 5.2. Notice that a subset E of R2 is countable Gδ if and only if it is scattered in R2. Indeed, assume that E is countable Gδ-set which contains a set Q without isolated points. Then Q is a Gδ-subset of R 2 which is homeomorphic to Q, a contradiction. On the other hand, if E is scattered, then Lemma 4.1 implies that E is countable. Since E is hereditarily Baire and countable, E is Gδ in R 2. Finally, we show that the Sorgenfrey plane is not a δ-normally separated space. Let E = {(x, −x) : x ∈ Q} and F = D \ E. Then E is closed and F is functionally closed in S2, since F is the difference of the functionally closed set D and the functionally open set ⋃ p∈E B[p, 1). But E and F can not be separated by disjoint clopen sets in S2, because E is not Gδ-subset of D in R 2. References [1] W. Bade, Two properties of the Sorgenfrey plane, Pacif. J. Math. 51, no. 2 (1974), 349–354. [2] R. Blair and A. Hager, Extensions of zero-sets and of real-valued functions, Math. Zeit. 136 (1974), 41–52. [3] G. Debs, Espaces héréditairement de Baire, Fund. Math. 129, no. 3 (1988), 199–206. [4] R. Engelking, General Topology. Revised and completed edition. Heldermann Verlag, Berlin (1989). [5] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton (1960). [6] R. Heath and E. Michael, A property of the Sorgenfrey line, Comp. Math. 23, no. 2 (1971), 185–188. [7] T. Hoshina and K. Yamazaki, Weak C-embedding and P -embedding, and product spaces, Topology Appl. 125 (2002), 233–247. [8] O. Kalenda and J. Spurný, Extending Baire-one functions on topological spaces, Topol- ogy Appl. 149 (2005), 195–216. [9] O. Karlova, On α-embedded sets and extension of mappings, Comment. Math. Univ. Carolin. 54, no. 3 (2013), 377–396. [10] J. Mack, Countable paracompactness and weak normality properties, Trans. Amer. Math. Soc. 148 (1970), 265–272. [11] H. Ohta, Extension properties and the Niemytzki plane, Appl. Gen. Topol. 1, no. 1 (2000), 45–60. [12] H. Ohta, K. Yamazaki, Extension problems of real-valued continuous functions, in: ”Open problems in topology II”, E. Pearl (ed.), Elsevier, 2007, 35–45. [13] J. Saint-Raymond, Jeux topologiques et espaces de Namioka, Proc. Amer. Math. Soc. 87, no. 3 (1983), 409–504. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 73 O. Karlova [14] W. Sierpiński, Sur une propriete topologique des ensembles denombrables denses en soi, Fund. Math. 1 (1920), 11–16. [15] Y. Tanaka, On closedness of C- and C∗-embeddings, Pacif. J. Math. 68, no. 1 (1977), 283–292. [16] J. Terasawa, On the zero-dimensionality of some non-normal product spaces, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 11 (1972), 167–174. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 74