TY - JOUR

T1 - Equalities for the extent of infinite products and Σ-products

AU - Hirata, Yasushi

AU - Usuba, Toshimichi

AU - Yajima, Yukinobu

N1 - Funding Information:
This research was supported by Grant-in-Aid for Scientific Research (C) 19K03606 , 18K03403 , 18K03404 .
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2022/2/15

Y1 - 2022/2/15

N2 - For a space X, let e(X)=ω⋅sup{|D|:D is a closed discrete subset in X}, which is called the extent of X. Here we deal with the following two questions: (1) For a product space X=∏λ∈ΛXλ, when is e(X)=|Λ|⋅sup{e(Xλ):λ∈Λ}? (2) For a Σ-product Σ of spaces Xλ,λ∈Λ, when is e(Σ)=sup{e(Xλ):λ∈Λ}? We show that the equalities in these questions hold if each Xλ is a strict p-space or a strong Σ-space and, in the case of the first question, if the cardinality of the index set Λ is less than the first weakly inaccessible. For semi-stratifiable spaces, we show that a slightly weaker form of these equalities holds.

AB - For a space X, let e(X)=ω⋅sup{|D|:D is a closed discrete subset in X}, which is called the extent of X. Here we deal with the following two questions: (1) For a product space X=∏λ∈ΛXλ, when is e(X)=|Λ|⋅sup{e(Xλ):λ∈Λ}? (2) For a Σ-product Σ of spaces Xλ,λ∈Λ, when is e(Σ)=sup{e(Xλ):λ∈Λ}? We show that the equalities in these questions hold if each Xλ is a strict p-space or a strong Σ-space and, in the case of the first question, if the cardinality of the index set Λ is less than the first weakly inaccessible. For semi-stratifiable spaces, we show that a slightly weaker form of these equalities holds.

KW - Extent

KW - p-Space

KW - Product

KW - Semi-stratifiable space

KW - Strict p-space

KW - Strong β-space

KW - Strong Σ-space

KW - Σ-product

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U2 - 10.1016/j.topol.2021.107946

DO - 10.1016/j.topol.2021.107946

M3 - Article

AN - SCOPUS:85122163232

VL - 307

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

M1 - 107946

ER -