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Abstract. We show that Sobczyk's Theorem holds for a new class of Banach spaces, namely spaces of continuous functions on linearly ordered compacta.

We obtain a new class of compact Hausdorff spaces K for which c0 can be nontrivially twisted with C(K). IntroductionIn this article, we present a broad new class of compact Hausdorff spaces K such that there exists a nontrivial twisted sum of c 0 and C(K), where C(K) denotes the Banach space of continuous real-valued functions on K endowed with the supremum norm. By a twisted sum of the Banach spaces Y and X we mean a short exact sequence 0 → Y → Z → X → 0, where Z is a Banach space and the maps are bounded linear operators. This twisted sum is called trivial if the exact sequence splits, i.e., if the map Y → Z admits a bounded linear left inverse (equivalently, if the map Z → X admits a bounded linear right inverse). In other words, the twisted sum is trivial if the range of the map Y → Z is complemented in Z; in this case, Z ∼ = X ⊕ Y . As in [7], we denote by Ext(X, Y ) the set of equivalence classes of twisted sums of Y and X and we write Ext(X, Y ) = 0 if every such twisted sum is trivial.Many problems in Banach space theory are related to the quest for conditions under which Ext(X, Y ) = 0. For instance, an equivalent statement for the classical Theorem of Sobczyk ([5, 13]) is that if X is a separable Banach space, then Ext(X, c 0 ) = 0 ([3, Proposition 3.2]). The converse of the latter statement clearly does not hold in general: for example, Ext ℓ 1 (I), c 0 = 0, since ℓ 1 (I) is a projective Banach space. However, the following question remains open: is it true that Ext C(K), c 0 = 0 for any nonseparable C(K) space? This problem was stated in [4,5] and further studied in the recent article [6], in which the author proves that, under the continuum hypothesis (CH), the space Ext C(K), c 0 is nonzero for a nonmetrizable compact Hausdorff space K of finite height. In addition to this result, everything else that is known about the problem is summarized in [6, Proposition 2], namely that Ext C(K), c 0 is nonzero for a C(K) space under any one of the following assumptions:

We present a characterization of the continuous increasing surjections φ : K → L between compact lines K and L for which the corresponding subalgebra φ * C(L) has the c0-extension property in C(K). A natural question arising in connection with this characterization is shown to be independent of the axioms of ZFC.Date: March 3rd, 2014. 2010 Mathematics Subject Classification. 46B20,46E15,54F05. Key words and phrases. Banach spaces of continuous functions; extensions of bounded operators; compact lines.The first author is sponsored by FAPESP (Process no. 2012/25171-0).

This cross-sectional study counted with the participation of 301 pregnant women seen in 2009 at a philanthropic maternity hospital in the city of São Paulo (a prenatal support program named Pré-Natal do Amparo Maternal - PN-AM). The objectives of this study were to evaluate the prenatal care according to the initial gestational age, the number of appointments that were held, the continuity of the assistance, and relate the appropriateness with the socio-demographic, obstetric and local variables of the initial prenatal care. The analysis criteria used was initiating prenatal care before 120 days of gestation and attending at least six appointments. The relationship between the variables was analyzed using the Chi-Square Test. Results showed that 41.5% of the pregnant women initiated prenatal care at another health care service and transferred spontaneously to the PN-AM; 74.1% initiated the prenatal care early and 80.4% attended at least six appointments; 63.1% met both criteria simultaneously. Appropriate prenatal care showed a statistically significant difference for mother's age, steady partner, employment, place of residence, having a companion during the appointment and place where prenatal care was initiated.