Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
$k$-Baire; almost $k$-discrete; point-$k$; locally-$k$
$k$-Baire; almost $k$-discrete; point-$k$; locally-$k$
Summary:
In this note we show the following theorem: ``Let $X$ be an almost $k$-discrete space, where $k$ is a regular cardinal. Then $X$ is $k^+$-Baire iff it is a $k$-Baire space and every point-$k$ open cover $\Cal U$ of $X$ such that $\operatorname{card}\, (\Cal U)\leq k$ is locally-$k$ at a dense set of points.'' For $k=\aleph _0$ we obtain a well-known characterization of Baire spaces. The case $k=\aleph _1$ is also discussed.
References:
[1] Fletcher P., Lindgren W.F.: A note on spaces of second category. Arch. der Math. 24 (1973), 186-187. MR 0315663 | Zbl 0259.54019
[2] Fogelgren J.R., McCoy R.A.: Some topological properties defined by homeomorphism groups. Arch. der Math. 22 (1971), 528-533. MR 0300259 | Zbl 0245.54018
[3] Haworth R.C., McCoy R.A.: Baire spaces. Dissertationes Math. 141 (1977), 1-73. MR 0431104 | Zbl 0344.54001
[4] Levy R.: Almost $P$-spaces. Can. J. Math. 29 (1977), 284-288. MR 0464203 | Zbl 0342.54032
[5] Tall F.D.: The countable chain condition versus separability - applications of Martin's axiom. Gen. Top. and Appl. 4 (1974), 315-339. MR 0423284 | Zbl 0293.54003
[6] Walker R.C.: The Stone-Čech compactification. Springer-Verlag, 1974. MR 0380698 | Zbl 0292.54001
[7] Weiss W.: Versions of Martin's axiom. in ``Handbook of Set-Theoretic Topology'', (K. Kunen and J.E. Vaughan, eds.), Elsevier Science Publishers, B.V., North Holland, 1984, pp. 827-886. MR 0776638 | Zbl 0571.54005
Search
Partner of
