Another note on the class of paracompact spaces whose product with every paracompact space is paracompact

The paper contains the following two results: X is a paracompact space and M is a metric space such that X can be embedded in M ω 1 in such a way that the projections of X onto initial countably many coordinates are closed, then the product X × Y is paracompact for every paracompact space Y if and only if the first player of the G ( D C , X ) game, introduced by Telgarsky, see Telgarsky (1975) [22], has a winning strategy. X is paracompact space, Y is a closed image of X and the first player of the G ( D C , X ) game has a winning strategy then also the first player of the G ( D C , Y ) game has a winning strategy.