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We prove that the $\sigma$-product of $\mathfrak E_{\mathrm{c}}$ ishomeomorphic to $\mathbb Q \times \mathfrak E_{\mathrm{c}}$, and construct aspace in $\mathrm{CAP}(\mathfrak E_{\mathrm{c}})$ which is not homeomorphic to$\mathbb Q \times \mathfrak E_{\mathrm{c}}$.

The $\sigma$-product and $\mathrm{CAP}$ of complete Erd\H{o}s space