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 C p ( Y , M ) admits a continuous injection into C p ( τ , M ) for some ordinal τ, then Y ¯ ∖ Y is hereditarily paracompact. This generalizes some known conclusions. If C p ( Y , M ) admits a continuous injection into C p ( Z , M ) for a separable space Z, then Y ¯ ∖ Y is hereditarily paracompact. If C p ( Y , M ) admits a continuous injection into C p ( L , M ) for a linearly ordered compactum L which satisfies that 1- c f ( min ⁡ L ) ≥ ω 1 , 0- c f ( max ⁡ L ) ≥ ω 1 , and i- c f ( x ) ≥ ω 1 for x ∈ L ∖ { max ⁡ L , min ⁡ L } and for i ∈ { 0 , 1 } , then Y ¯ ∖ Y is hereditarily paracompact.