A note on condensations of function spaces onto $σ$-compact and analytic spaces

Modifying a construction of W. Marciszewski we prove (in ZFC) that there exists a subspace of the real line $\mathbb{R}$, such that the realcompact space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology does not admit a continuous bijection onto a $σ$-compact space. This answers a question of Arhangel'skii.

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