A note on the Menger (Rothberger) property and topological games

In this note we show that a T1T1 topological space X   is a Menger space if and only if for each sequence {ϕn:n∈ω}{ϕn:n∈ω} of neighborhood assignments for X  , there exists, for each n∈ωn∈ω, a finite subset DnDn of X   such that X=⋃{ϕn(d):d∈Dn,n∈ω}X=⋃{ϕn(d):d∈Dn,n∈ω} and D=⋃{Dn:n∈ω}D=⋃{Dn:n∈ω} is a closed discrete subspace of X  . A T1T1 topological space X   is a Rothberger space if and only if for each sequence {ϕn:n∈ω}{ϕn:n∈ω} of neighborhood assignments for X  , there exists, for each n∈ωn∈ω, a point dn∈Xdn∈X such that X=⋃{ϕn(dn):n∈ω}X=⋃{ϕn(dn):n∈ω} and D={dn:n∈ω}D={dn:n∈ω} is a closed discrete subspace of X.Let M be the class of all spaces with the Menger property. Let R be the class of all Rothberger spaces. We show that a M-like space has the Menger property. A R-like space is a Rothberger space.Let C be the class of all compact spaces. Let W be the class of all countable spaces. We prove that a finite product of C-like spaces is a C-like space. As a corollary we know that a finite product of C-like spaces is a Menger space. In the last part of this note we show that a finite product of W-like Hausdorff spaces is a nc-W-like space. A finite product of W-like spaces is a Rothberger space.