More on injections into function spaces over ordinals (Q408564)

For topological spaces \(X\) and \(Y\) let \(C_p(X,Y)\) denote the space of all continuous functions from \(X\) into \(Y\) endowed with the topology of pointwise convergence. Let \(X\) be a subspace of an ordinal \(\lambda\) and \(M\) be a non-trivial metric space. If \(C_p(X,M)\) admits a continuous injection into \(C_p(\tau, M)\) for some ordinal \(\tau\) then no subspace of \(\overline{X}\setminus X\) is homeomorphic to a stationary subspace of an uncountable regular ordinal \(\kappa\), so \(\overline{X}\setminus X\) is hereditarily paracompact.