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A space $X$ is callled semi-stratifiable space if it has a $g$-function such that: for any point $x$ of $X$ and a sequence $\{x_n\}$ of $X$ if $x \in g(n,x_n)$, then $x_n \to x$.

Note that every Moore space is semi-stratifiable. We know the cardinality of a star countable Moore space is not greater than $\mathfrak c$.

A topological space $X$ is said to be star countable if whenever $\mathscr{U}$ is an open cover of $X$, there is a countable subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$.

Is there a star countable semi-stratifiable space $X$ with $|X|> \mathfrak c$?

Thanks for your help.

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Could you kindly tell us what a $g$-function is? –  Joel David Hamkins May 29 '13 at 11:32
    
$g: \mathbb N \times X \to \tau_X$ is a $g$-function of $X$ if for any $x$ and $n \in \mathbb N$, $x \in g(n+1,x) \subset g(n,x)$. –  Paul May 29 '13 at 11:40
    
It seems that you want to impose some separation axiom, since otherwise the indiscrete space (of any cardinality) would seem to be trivially semi-stratifiable and star-countable. –  Joel David Hamkins May 29 '13 at 15:00
    
It's customary in generalised metric spaces to assume at least $T_3$ ($T_1$ plus regular). This is also customary for stratifiable and semi-stratifiable spaces, AFAIK. –  Henno Brandsma May 29 '13 at 18:12
    
@Joel: maybe I should mentioned it. –  Paul May 30 '13 at 0:03

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