SOME WEAKER FORMS OF THE CHAIN (F) CONDITION FOR METACOMPACTNESS


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Vural Ç.

JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, cilt.84, sa.2, ss.283-288, 2008 (SCI İndekslerine Giren Dergi) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 84 Konu: 2
  • Basım Tarihi: 2008
  • Doi Numarası: 10.1017/s1446788708000037
  • Dergi Adı: JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY
  • Sayfa Sayıları: ss.283-288

Özet

We define, in a slightly unusual way, the rank of a partially ordered set. Then we prove that if X is a topological space and W = {W(x) : x is an element of X} satisfies condition (F) and, for every x is an element of X, W(x) is of the form boolean OR(i is an element of n(x)) W(i)(x), where W(0)(x) is Noetherian of finite rank, and every other W(i) (x) is a chain (with respect to inclusion) of neighbourhoods of x, then X is metacompact. We also obtain a cardinal extension of the above. In addition, we give a new proof of the theorem 'if the space X has a base B of point-finite rank, then X is metacompact', which was proved by Gruenhage and Nyikos.