Injections into function spaces over ordinals and LOTS

To formulate our results let M be a metric space with at least two points and let Y be a subspace of a generalized ordered (GO) space X  . We get the following conclusions: If Cp(Y,M)Cp(Y,M) admits a continuous injection into Cp(τ,M)Cp(τ,M) for some ordinal τ  , then Y¯∖Y is hereditarily paracompact. This generalizes some known conclusions. If Cp(Y,M)Cp(Y,M) admits a continuous injection into Cp(Z,M)Cp(Z,M) for a separable space Z  , then Y¯∖Y is hereditarily paracompact. If Cp(Y,M)Cp(Y,M) admits a continuous injection into Cp(L,M)Cp(L,M) for a linearly ordered compactum L   which satisfies that 1-cf(min⁡L)≥ω1cf(min⁡L)≥ω1, 0-cf(max⁡L)≥ω1cf(max⁡L)≥ω1, and i  -cf(x)≥ω1cf(x)≥ω1 for x∈L∖{max⁡L,min⁡L}x∈L∖{max⁡L,min⁡L} and for i∈{0,1}i∈{0,1}, then Y¯∖Y is hereditarily paracompact.