Answering a question on relative countable paracompactness

In [6], Yoshikazu Yasui formulates some results on relative countable paracompactness and poses some questions. Like it is the case with many other topological properties [1], countable paracompactness has several possible relativizations. Thus a subspace Y ⊂ X is called countably 1-paracompact in X provided for every countable open cover U of X there is an open cover V of X which refines U and is locally finite at the points of Y (i.e. every point of Y has a neighbourhood in X that meets only finitely many elements of V). Yasui asserts that if a countably compact space Y is closed in a normal space X then Y is countably 1-paracompact in X and asks (Problem 1 in[6]) if normality can be omitted. The answer is negative as it is demonstrated by X = ((ω1 + 1)× (ω + 1)) \ {(ω1, ω)}, Y = ω1 × {ω} and U = {ω1 × (ω + 1)} ∪ {{(ω1 + 1) × {n}} : n ∈ ω}. This well-known construction provides also the following general statement (recall that a space is Linearly Lindelöf iff every uncountable set of regular cardinality has a complete accumulation point see e.g. [2]).