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

The paper contains the following two results:(1) If X is a paracompact space and M is a metric space such that X can be embedded in Mω1 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(DC,X) game, introduced by Telgarsky, see Telgarsky (1975) [22], has a winning strategy.(2) If X is paracompact space, Y is a closed image of X and the first player of the G(DC,X) game has a winning strategy then also the first player of the G(DC,Y) game has a winning strategy.