For a space X, let e(X)=ω⋅sup{|D|:D is closed discrete subset in X}, which called the extent of X. Here we deal with following two questions: product X=∏λ∈ΛXλ, when e(X)=|Λ|⋅sup{e(Xλ):λ∈Λ}? Σ-product Σ spaces Xλ,λ∈Λ, e(Σ)=sup{e(Xλ):λ∈Λ}?
Let $mathcal{H}$ and $mathcal{K}$ be infinite dimensional Hilbert spaces, while $mathcal{B(H)}$ and $mathcal{B(K)}$ denote the algebras of all linear bounded operators on $mathcal{H}$ and $mathcal{K}$, respectively. We characterize the forms of additive mappings from $mathcal{B(H)}$ into $mathcal{B(K)}$ that preserve the nonzero idempotency of either Jordan products of operators or usual produc...
