A note on condensations of function spaces onto $\sigma$-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 $\sigma$-compact space. This answers a question of Arhangel’skii.