TY - JOUR

T1 - Free locally convex spaces with a small base

AU - Gabriyelyan, Saak

AU - Ka̧kol, Jerzy

N1 - Funding Information:
The research was supported for Jerzy Ka̧kol by Generalitat Valenciana, Conselleria d’Educació i Esport, Spain, Grant PROMETEO/2015/058 and by the GAČR project 16-34860L and RVO: 67985840. Jerzy Ka̧kol gratefully acknowledges also the financial support he received from the Center for Advanced Studies in Mathematics of the Ben Gurion University of the Negev during his visit March 15–22, 2016.
Publisher Copyright:
© 2016, Springer-Verlag Italia.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - The paper studies the free locally convex space L(X) over a Tychonoff space X. Since for infinite X the space L(X) is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for L(X) is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a G-base. A space X has a G-base if for every x∈ X there is a base { Uα: α∈ NN} of neighborhoods at x such that Uβ⊆ Uα whenever α≤ β for all α, β∈ NN, where α= (α(n)) n ∈ N≤ β= (β(n)) n ∈ N if α(n) ≤ β(n) for all n∈ N. We show that if X is an Ascoli σ-compact space, then L(X) has a G-base if and only if X admits an Ascoli uniformity U with a G-base. We prove that if X is a σ-compact Ascoli space of NN-uniformly compact type, then L(X) has a G-base. As an application we show: (1) if X is a metrizable space, then L(X) has a G-base if and only if X is σ-compact, and (2) if X is a countable Ascoli space, then L(X) has a G-base if and only if X has a G-base.

AB - The paper studies the free locally convex space L(X) over a Tychonoff space X. Since for infinite X the space L(X) is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for L(X) is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a G-base. A space X has a G-base if for every x∈ X there is a base { Uα: α∈ NN} of neighborhoods at x such that Uβ⊆ Uα whenever α≤ β for all α, β∈ NN, where α= (α(n)) n ∈ N≤ β= (β(n)) n ∈ N if α(n) ≤ β(n) for all n∈ N. We show that if X is an Ascoli σ-compact space, then L(X) has a G-base if and only if X admits an Ascoli uniformity U with a G-base. We prove that if X is a σ-compact Ascoli space of NN-uniformly compact type, then L(X) has a G-base. As an application we show: (1) if X is a metrizable space, then L(X) has a G-base if and only if X is σ-compact, and (2) if X is a countable Ascoli space, then L(X) has a G-base if and only if X has a G-base.

KW - C(X)

KW - Compact resolution

KW - Free locally convex space

KW - G-base

UR - http://www.scopus.com/inward/record.url?scp=85015383055&partnerID=8YFLogxK

U2 - 10.1007/s13398-016-0315-1

DO - 10.1007/s13398-016-0315-1

M3 - Article

AN - SCOPUS:85015383055

SN - 1578-7303

VL - 111

SP - 575

EP - 585

JO - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

JF - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

IS - 2

ER -