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1. CMB Online first
Topological games and Alster spaces In this paper we study connections between topological games
such
as Rothberger, Menger and compact-open, and relate these games
to
properties involving covers by $G_\delta$ subsets. The results
include:
(1) If Two has a winning strategy in the Menger
game on a regular space $X$, then $X$ is an Alster space.
(2) If Two has a winning strategy in the Rothberger game on a
topological space $X$, then the $G_\delta$-topology on $X$ is
LindelÃ¶f.
(3) The Menger game and the compact-open game are (consistently)
not
dual.
Keywords:topological games, selection principles, Alster spaces, Menger spaces, Rothberger spaces, Menger game, Rothberger game, compact-open game, $G_\delta$-topology Categories:54D20, 54G99, 54A10 |
2. CMB Online first
On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces We establish that if it is consistent that there is a
supercompact cardinal, then it is consistent that every locally
compact, hereditarily normal space which does not include a perfect
pre-image of $\omega_1$ is hereditarily paracompact.
Keywords:locally compact, hereditarily normal, paracompact, Axiom R, PFA$^{++}$ Categories:54D35, 54D15, 54D20, 54D45, 03E65, 03E35 |
3. CMB 2011 (vol 56 pp. 203)
Productively LindelÃ¶f Spaces May All Be $D$ We give easy proofs that (a) the Continuum Hypothesis implies that if
the product of $X$ with every LindelÃ¶f space is LindelÃ¶f, then $X$ is
a $D$-space, and (b) Borel's Conjecture implies every Rothberger space
is Hurewicz.
Keywords:productively LindelÃ¶f, $D$-space, projectively $\sigma$-compact, Menger, Hurewicz Categories:54D20, 54B10, 54D55, 54A20, 03F50 |
4. CMB 1998 (vol 41 pp. 245)
The normality in products with a countably compact factor It is known that the product $\omega_1 \times X$ of
$\omega_1$ with an $M_1$-space may be nonnormal. In this paper we
prove that the product $\kappa \times X$ of an uncountable regular
cardinal $\kappa$ with a paracompact semi-stratifiable space is normal
if{f} it is countably paracompact. We also give a sufficient
condition under which the product of a normal space with a paracompact
space is normal, from which many theorems involving such a product
with a countably compact factor can be derived.
Categories:54B19, 54D15, 54D20 |
5. CMB 1997 (vol 40 pp. 395)
$D$-spaces and resolution A space $X$ is a $D$-space if, for every neighborhood assignment $f$
there is a closed discrete set $D$ such that $\bigcup{f(D)}=X$. In this
paper we give some necessary conditions and some sufficient conditions
for a resolution of a topological space to be a $D$-space. In particular,
if a space $X$ is resolved at each $x\in X$ into a $D$-space $Y_x$ by
continuous mappings $f_x\colon X-\{{x}\} \rightarrow Y_x$, then the
resolution is a $D$-space if and only if $\bigcup{\{{x}\}}\times \Bd(Y_x)$
is a $D$-space.
Keywords:$D$-space, neighborhood assignment, resolution, boundary Categories:54D20, 54B99, 54D10, 54D30 |