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

Let $X$ have countable extent and is locally compact. Then must $X^2$ be star $\sigma$-compact?

Thanks for your help.

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