# Is $\Box_{n\in\omega} \mathbb{R}$ metacompact?

Is $\Box_{n\in\omega} \mathbb{R}$ (that is $\mathbb{R}^\omega$ endowed with the box topology) metacompact?

• Wikipedia says that the box topology on $\mathbb{R}^\omega$ is paracompact if the continuum hypothesis holds and paracompact spaces are metacompact. So under CH the answer is positive. – James Hanson Aug 21 '18 at 16:29
• It seems that the paracompactness of the box topology on $\mathbb R^\omega$ can be proved under $\mathfrak b=\mathfrak d$ (which is equivalent to the existence of a scale in $\omega^\omega$). – Taras Banakh Aug 21 '18 at 20:44
• @TarasBanakh, Also under $\mathfrak{b}=\aleph_1$ and under $\mathfrak{d}=\mathfrak{c}$. – Ramiro de la Vega Aug 21 '18 at 22:30