Applied General Topology - Vol 06, No 1 (2005)

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Tabla de contenidos



  • On functionally θ-normal spaces
  • The character of free topological groups I
  • The character of free topological groups II
  • Abelization of join spaces of affine transformations of ordered field with proximity
  • δ-closure, θ-closure and generalized closed sets
  • A generalized coincidence point index
  • Remarks on the finite derived set property

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Browsing Applied General Topology - Vol 06, No 1 (2005) by Subject "Compact"

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    The character of free topological groups I
    (Universitat Politècnica de València, 2005-04-01) Nickolas, Peter; Tkachenko, Mikhail; Consejo Nacional de Ciencia y Tecnología, México
    [EN] A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself. From these results, others follow on the basis of various topological assumptions. Amongst these: (i) if X is a compact Hausdorff space, then the character of the free abelian topological group on X lies between w(X) and w(X)ℵ0, where w(X) denotes the weight of X; (ii) if the Tychonoff space X is not a P-space, then the character of the free abelian topological group is bounded below by the “small cardinal” d; and (iii) if X is an infinite compact metrizable space, then the character is precisely d. In the non-abelian case, we show that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group, but the inequality is in general strict. It is also shown that the characters of the free abelian and the free topological groups are equal whenever the given uniform space is w-narrow. A sequel to this paper analyses more closely the cases of the free and free abelian topological groups on compact Hausdorff spaces and metrizable spaces.
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    The character of free topological groups II
    (Universitat Politècnica de València, 2005-04-01) Nickolas, Peter; Tkachenko, Mikhail; Consejo Nacional de Ciencia y Tecnología, México
    [EN] A systematic analysis is made of the character of the free and free abelian topological groups on metrizable spaces and compact spaces, and on certain other closely related spaces. In the first case, it is shown that the characters of the free and the free abelian topological groups on X are both equal to the “small cardinal” d if X is compact and metrizable, but also, more generally, if X is a non-discrete k!-space all of whose compact subsets are metrizable, or if X is a non-discrete Polish space. An example is given of a zero-dimensional separable metric space for which both characters are equal to the cardinal of the continuum. In the case of a compact space X, an explicit formula is derived for the character of the free topological group on X involving no cardinal invariant of X other than its weight; in particular the character is fully determined by the weight in the compact case. This paper is a sequel to a paper by the same authors in which the characters of the free groups were analysed under less restrictive topological assumptions.