34 found
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  1.  5
    The Existence of Free Ultrafilters on Ω Does Not Imply the Extension of Filters on Ω to Ultrafilters.Eric J. Hall, Kyriakos Keremedis & Eleftherios Tachtsis - 2013 - Mathematical Logic Quarterly 59 (4-5):258-267.
  2.  19
    Products of Some Special Compact Spaces and Restricted Forms of AC.Kyriakos Keremedis & Eleftherios Tachtsis - 2010 - Journal of Symbolic Logic 75 (3):996-1006.
    We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1] (...)
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  3.  40
    Properties of the Real Line and Weak Forms of the Axiom of Choice.Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Eleftherios Tachtsis - 2005 - Mathematical Logic Quarterly 51 (6):598-609.
    We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals.
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  4.  18
    Some Weak Forms of the Axiom of Choice Restricted to the Real Line.Kyriakos Keremedis & Eleftherios Tachtsis - 2001 - Mathematical Logic Quarterly 47 (3):413-422.
    It is shown that AC, the axiom of choice for families of non-empty subsets of the real line ℝ, does not imply the statement PW, the powerset of ℝ can be well ordered. It is also shown that the statement “the set of all denumerable subsets of ℝ has size 2math image” is strictly weaker than AC and each of the statements “if every member of an infinite set of cardinality 2math image has power 2math image, then the union has (...)
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  5.  18
    The Compactness of 2^R and the Axiom of Choice.Kyriakos Keremedis - 2000 - Mathematical Logic Quarterly 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.
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  6.  41
    Products of Compact Spaces and the Axiom of Choice II.Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin - 2003 - Mathematical Logic Quarterly 49 (1):57-71.
    This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.
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  7.  32
    Metric Spaces and the Axiom of Choice.Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin - 2003 - Mathematical Logic Quarterly 49 (5):455-466.
    We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.
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  8.  4
    Tychonoff Products of Compact Spaces in ZF and Closed Ultrafilters.Kyriakos Keremedis - 2010 - Mathematical Logic Quarterly 56 (5):474-487.
    Let {: i ∈I } be a family of compact spaces and let X be their Tychonoff product. [MATHEMATICAL SCRIPT CAPITAL C] denotes the family of all basic non-trivial closed subsets of X and [MATHEMATICAL SCRIPT CAPITAL C]R denotes the family of all closed subsets H = V × Πmath imageXi of X, where V is a non-trivial closed subset of Πmath imageXi and QH is a finite non-empty subset of I. We show: Every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL (...)
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  9.  50
    Unions and the Axiom of Choice.Omar De la Cruz, Eric J. Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin - 2008 - Mathematical Logic Quarterly 54 (6):652-665.
    We study statements about countable and well-ordered unions and their relation to each other and to countable and well-ordered forms of the axiom of choice. Using WO as an abbreviation for “well-orderable”, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union (...)
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  10.  22
    The Vector Space Kinna-Wagner Principle is Equivalent to the Axiom of Choice.Kyriakos Keremedis - 2001 - Mathematical Logic Quarterly 47 (2):205-210.
    We show that the axiom of choice AC is equivalent to the Vector Space Kinna-Wagner Principle, i.e., the assertion: “For every family [MATHEMATICAL SCRIPT CAPITAL V]= {Vi : i ∈ k} of non trivial vector spaces there is a family ℱ = {Fi : i ∈ k} such that for each i ∈ k, Fiis a non empty independent subset of Vi”. We also show that the statement “every vector space over ℚ has a basis” implies that every infinite well (...)
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  11.  6
    Partition Reals and the Consistency of T > Add(R).Kyriakos Keremedis - 1993 - Mathematical Logic Quarterly 39 (1):545-550.
    We show that it is consistent with ZFC that the additivity number add of the ideal of meager sets of the real line is strictly greater than the tower number t of the reals. MSC: 03E35, 54D20.
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  12.  18
    Disjoint Unions of Topological Spaces and Choice.Paul Howard, Kyriakos Keremedis, Herman Rubin & Jean E. Rubin - 1998 - Mathematical Logic Quarterly 44 (4):493-508.
    We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice.
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  13.  13
    The Failure of the Axiom of Choice Implies Unrest in the Theory of Lindelöf Metric Spaces.Kyriakos Keremedis - 2003 - Mathematical Logic Quarterly 49 (2):179-186.
    In the realm of metric spaces the role of choice principles is investigated.
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  14.  19
    Extending Independent Sets to Bases and the Axiom of Choice.Kyriakos Keremedis - 1998 - Mathematical Logic Quarterly 44 (1):92-98.
    We show that the both assertions “in every vector space B over a finite element field every subspace V ⊆ B has a complementary subspace S” and “for every family [MATHEMATICAL SCRIPT CAPITAL A] of disjoint odd sized sets there exists a subfamily ℱ={Fj:j ϵω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ℚ every generating set includes a basis”.
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  15.  9
    On Sequentially Compact Subspaces Of.Kyriakos Keremedis & Eleftherios Tachtsis - 2003 - Notre Dame Journal of Formal Logic 44 (3):175-184.
    We show that the property of sequential compactness for subspaces of.
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  16.  7
    On Sequentially Compact Subspaces of Without the Axiom of Choice.Kyriakos Keremedis & Eleftherios Tachtsis - 2003 - Notre Dame Journal of Formal Logic 44 (3):175-184.
    We show that the property of sequential compactness for subspaces of.
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  17.  35
    Disasters in Topology Without the Axiom of Choice.Kyriakos Keremedis - 2001 - Archive for Mathematical Logic 40 (8):569-580.
    We show that some well known theorems in topology may not be true without the axiom of choice.
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  18.  10
    Powers of 2.Kyriakos Keremedis & Horst Herrlich - 1999 - Notre Dame Journal of Formal Logic 40 (3):346-351.
    It is shown that in ZF Martin's -axiom together with the axiom of countable choice for finite sets imply that arbitrary powers 2X of a 2-point discrete space are Baire; and that the latter property implies the following: (a) the axiom of countable choice for finite sets, (b) power sets of infinite sets are Dedekind-infinite, (c) there are no amorphous sets, and (d) weak forms of the Kinna-Wagner principle.
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  19.  26
    A Note on Shoenfield's Unramified Forcing.Kyriakos Keremedis - 1991 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 37 (9-12):183-186.
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  20.  6
    On Lindelof Metric Spaces and Weak Forms of the Axiom of Choice.Kyriakos Keremedis & Eleftherios Tachtsis - 2000 - Mathematical Logic Quarterly 46 (1):35-44.
    We show that the countable axiom of choice CAC strictly implies the statements “Lindelöf metric spaces are second countable” “Lindelöf metric spaces are separable”. We also show that CAC is equivalent to the statement: “If is a Lindelöf topological space with respect to the base ℬ, then is Lindelöf”.
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  21.  4
    Powers Of.Kyriakos Keremedis & Horst Herrlich - 1999 - Notre Dame Journal of Formal Logic 40 (3):346-351.
    It is shown that in ZF Martin's $ \aleph_{0}^{}$-axiom together with the axiom of countable choice for finite sets imply that arbitrary powers 2X of a 2-point discrete space are Baire; and that the latter property implies the following: the axiom of countable choice for finite sets, power sets of infinite sets are Dedekind-infinite, there are no amorphous sets, and weak forms of the Kinna-Wagner principle.
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  22.  29
    Filters, Antichains and Towers in Topological Spaces and the Axiom of Choice.Kyriakos Keremedis - 1998 - Mathematical Logic Quarterly 44 (3):359-366.
    We find some characterizations of the Axiom of Choice in terms of certain families of open sets in T1 spaces.
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  23.  26
    Versions of Normality and Some Weak Forms of the Axiom of Choice.Paul Howard, Kyriakos Keremedis, Herman Rubin & Jean E. Rubin - 1998 - Mathematical Logic Quarterly 44 (3):367-382.
    We investigate the set theoretical strength of some properties of normality, including Urysohn's Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces.
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  24.  4
    Weak Hausdorff Gaps and The.Kyriakos Keremedis - 1999 - Mathematical Logic Quarterly 45 (1):95-104.
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  25.  21
    Some Remarks on Category of the Real Line.Kyriakos Keremedis - 1999 - Archive for Mathematical Logic 38 (3):153-162.
    We find a characterization of the covering number $cov({\mathbb R})$ , of the real line in terms of trees. We also show that the cofinality of $cov({\mathbb R})$ is greater than or equal to ${\mathfrak n}_\lambda$ for every $\lambda \in cov({\mathbb R}),$ where $\mathfrak n_\lambda \geq add({\mathcal L})$ ( $add( {\mathcal L})$ is the additivity number of the ideal of all Lebesgue measure zero sets) is the least cardinal number k for which the statement: $(\exists{\mathcal G}\in [^\omega \omega ]^{\leq \lambda (...)
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  26.  10
    Non-Discrete Metrics in and Some Notions of Finiteness.Kyriakos Keremedis - 2016 - Mathematical Logic Quarterly 62 (4-5):383-390.
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  27.  19
    Nonconstructive Properties of Well-Ordered T 2 Topological Spaces.Kyriakos Keremedis & Eleftherios Tachtsis - 1999 - Notre Dame Journal of Formal Logic 40 (4):548-553.
    We show that none of the following statements is provable in Zermelo-Fraenkel set theory (ZF) answering the corresponding open questions from Brunner in ``The axiom of choice in topology'':(i) For every T2 topological space (X, T) if X is well-ordered, then X has a well-ordered base,(ii) For every T2 topological space (X, T), if X is well-ordered, then there exists a function f : X × W T such that W is a well-ordered set and f ({x} × W) is (...)
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  28.  15
    Countable Sums and Products of Metrizable Spaces in ZF.Kyriakos Keremedis & Eleftherios Tachtsis - 2005 - Mathematical Logic Quarterly 51 (1):95-103.
    We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, Lindelöf metric spaces, and in disjoint topological unions of countably many such spaces.
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  29.  4
    Two New Equivalents of Lindelöf Metric Spaces.Kyriakos Keremedis - 2018 - Mathematical Logic Quarterly 64 (1-2):37-43.
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  30.  8
    On Sequentially Closed Subsets of the Real Line in ZF.Kyriakos Keremedis - 2015 - Mathematical Logic Quarterly 61 (1-2):24-31.
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  31.  9
    The Boolean Prime Ideal Theorem and Products of Cofinite Topologies.Kyriakos Keremedis - 2013 - Mathematical Logic Quarterly 59 (6):382-392.
  32.  7
    Consequences of the Failure of the Axiom of Choice in the Theory of Lindelof Metric Spaces.Kyriakos Keremedis - 2004 - Mathematical Logic Quarterly 50 (2):141.
    We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: Every Lindelöf metric space is separable and Every Lindelöf metric space is second countable are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice restricted to countable sets and to (...)
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  33.  7
    A Note on Shoenfield's Unramified Forcing.Kyriakos Keremedis - 1991 - Mathematical Logic Quarterly 37 (9‐12):183-186.
  34.  6
    On Countable Products of Finite Hausdorff Spaces.Horst Herrlich & Kyriakos Keremedis - 2000 - Mathematical Logic Quarterly 46 (4):537-542.
    We investigate in ZF conditions that are necessary and sufficient for countable products ∏m∈ℕXm of finite Hausdorff spaces Xm resp. Hausdorff spaces Xm with at most n points to be compact resp. Baire. Typica results: Countable products of finite Hausdorff spaces are compact if and only if countable products of non-empty finite sets are non-empty. Countable products of discrete spaces with at most n + 1 points are compact if and only if countable products of non-empty sets with at most (...)
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