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43.
Kyriakos Keremedis 《Mathematical Logic Quarterly》2015,61(1-2):24-31
We show:
- (i) iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete.
- (ii) Every infinite subset X of has a countably infinite subset iff every infinite sequentially closed subset of includes an infinite closed subset.
- (iii) The statement “ is sequential” is equivalent to each one of the following propositions:
- (a) Every sequentially closed subset A of includes a countable cofinal subset C,
- (b) for every sequentially closed subset A of , is a meager subset of ,
- (c) for every sequentially closed subset A of , ,
- (d) every sequentially closed subset of is separable,
- (e) every sequentially closed subset of is Cantor complete,
- (f) every complete subspace of is Cantor complete.
44.
In ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice, the category Top of topological spaces and continuous maps is well-behaved. In particular, Top has sums (=coproducts) and products. However, it may happen that for families (Xi)i∈I and (Yi)i∈I with the property that each Xi is homeomorphic to the corresponding Yi neither their sums ⊕i∈IXi and ⊕i∈IYi nor their products ∏i∈IXi and ∏i∈IYi are homeomorphic. It will be shown that the axiom of choice is not only sufficient but also necessary to rectify this defect. 相似文献
45.
Kyriakos Keremedis 《Mathematical Logic Quarterly》2010,56(5):475-487
Let {(Xi, Ti): i ∈I } be a family of compact spaces and let X be their Tychonoff product. ??(X) denotes the family of all basic non‐trivial closed subsets of X and ??R(X) denotes the family of all closed subsets H = V × ΠXi of X, where V is a non‐trivial closed subset of ΠXi and QH is a finite non‐empty subset of I. We show: (i) Every filterbase ?? ? ??R(X) extends to a ??R(X)‐ultrafilter ? if and only if every family H ? ??(X) with the finite intersection property (fip for abbreviation) extends to a maximal ??(X) family F with the fip. (ii) The proposition “if every filterbase ?? ? ??R(X) extends to a ??R(X)‐ultrafilter ?, then X is compact” is not provable in ZF. (iii) The statement “for every family {(Xi, Ti): i ∈ I } of compact spaces, every filterbase ?? ? ??R(Y), Y = Πi ∈IYi, extends to a ??R(Y)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem. (iv) The statement “for every family {(Xi, Ti): i ∈ ω } of compact spaces, every countable filterbase ?? ? ??R(X), X = Πi ∈ωXi, extends to a ??R(X)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem restricted to countable families. (v) The countable Axiom of Choice is equivalent to the proposition “for every family {(Xi, Ti): i ∈ ω } of compact topological spaces, every countable family ?? ? ??(X) with the fip extends to a maximal ??(X) family ? with the fip” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
46.
A novel method for the synthesis of densely functionalized optically active 5-carboxymethyl tetramic acids is reported. The proposed strategy is focused on the synthesis of 3,3′-disubstituted pyrrolidine-2,4-diones as useful molecules for the synthesis of natural products. The proposed methodology has been established by X-ray diffraction structure analysis. 相似文献
47.
Jones MA Britz DA Morton JJ Khlobystov AN Porfyrakis K Ardavan A Briggs GA 《Physical chemistry chemical physics : PCCP》2006,8(17):2083-2088
The endohedral fullerene epoxide N@C60O was synthesised, isolated by High Performance Liquid Chromatography (HPLC), and characterised by Electron Spin Resonance (ESR). This nitrogen radical displays predominantly axial symmetry characteristics as expected for a monoadduct, evidenced by a zero-field splitting D parameter of 6.6 MHz and an E parameter of 0.5 MHz in powder at 77 K. Photo- and thermally-activated silencing of the nitrogen radical were observed, the latter showing the evolution of a new spin signal during heating at 100 degrees C. We suggest that loss of nitrogen spin is due to coupling with a radical formed by opening of the epoxide ring. This implies that the reaction of C60O with C60 in the solid state proceeds via a radical, rather than ionic, intermediate. 相似文献
48.
Wen L Cheng J Zou H Zhang L Chen J Papadopoulos KD 《Langmuir : the ACS journal of surfaces and colloids》2004,20(19):8391-8397
A mathematical model for analyzing the van der Waals interaction between the internal aqueous droplets (W(1)) and the external aqueous phase (W(2)) of double emulsions has been established. The effects of Hamaker constants of the materials forming the system, especially those of the two different adsorbed surfactant layers with uniform density (A(1) and A(2)), on the van der Waals interaction were investigated. The overall van der Waals interaction across the oil film is a combined result of four individual parts, that is, W(1)-W(2), A(1)-A(2), W(1)-A(1), and A(2)-W(2) van der Waals interaction, and it may be either attractive or repulsive depending on many factors. It was found that the overall van der Waals interaction is dominated by the W(1)-W(2) interaction at large separation distances between the W(1)/O and O/W(2) interfaces, while it is mostly determined by the A(1)-A(2) interaction when the two interfaces are extremely close. Specifically, in the cases when the value of the Hamaker constant of the oil phase is intermediate between those of W(1) and W(2) and there is a thick oil film separating the two interfaces, a weak repulsive overall van der Waals interaction will prevail. If the Hamaker constant of the oil phase is intermediate between those of A(1) and A(2) and the two interfaces are very close, the overall van der Waals interaction will be dominated by the strong repulsive A(1)-A(2) interaction. The repulsive van der Waals interaction at such cases helps stabilize the double emulsions. 相似文献
49.
Kyriakos Keremedis 《Mathematical Logic Quarterly》2000,46(4):569-571
We show that for every we ordered cardinal number m the Tychonoff product 2m is a compact space without the use of any choice but in Cohen's Second Mode 2ℝ is not compact. 相似文献
50.
It is shown that AC(ℝ), the axiom of choice for families of non‐empty subsets of the real line ℝ, does not imply the statement PW(ℝ), the powerset of ℝ can be well ordered. It is also shown that (1) the statement “the set of all denumerable subsets of ℝ has size 2 ” is strictly weaker than AC(ℝ) and (2) each of the statements (i) “if every member of an infinite set of cardinality 2 has power 2 , then the union has power 2 ” and (ii) “ℵ(2 ) ≠ ℵω” (ℵ(2 ) is Hartogs' aleph, the least ℵ not ≤ 2 ), is strictly weaker than the full axiom of choice AC. 相似文献