Hereditarily Lindelöf and hereditarily separable subspaces of function spaces
Abstract
A space is ESLC if it contains neither an S-space nor an L-space, and NFS is the class of spaces containing no free sequences of length ω1. We prove that under MA(ω1), if all finite powers of X belong to NFS, then Cp(X)ω is ESLC; thus, for subspaces of Cp(X), hereditary Lindelöfness, hereditary separability and countable spread coincide. This is derived from a dual form of a theorem of Todorčević. A Lindelöf Σ-space belongs to NFS if and only if its compact subspaces have countable tightness. Hence Cp(X) contains no L-spaces if X is a Lindelöf Σ-space of countable tightness and MA(ω1) holds, or if X is any Lindelöf Σ-space and PFA holds. Under MA(ω1), all iterated function spaces of Corson compact spaces are ESLC, and in ZFC the space Cp(X,{0,1}) is ESLC for every zero-dimensional Corson compact space X.
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BibTeX
@misc{reznichenko2026hereditarily,
title = {{Hereditarily Lindelöf and hereditarily separable subspaces of function spaces}},
author = {Evgenii Reznichenko},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04911},
eprint = {2611.04911},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04911},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hereditarily Lindelöf and hereditarily separable subspaces of function spaces
AU - Evgenii Reznichenko
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04911
UR - https://arxiv.org/abs/2611.04911
AB - A space is ESLC if it contains neither an S-space nor an L-space, and NFS is the class of spaces containing no free sequences of length ω_1. We prove that under MA(ω_1), if all finite powers of X belong to NFS, then C_p(X)^ω is ESLC; thus, for subspaces of C_p(X), hereditary Lindelöfness, hereditary separability and countable spread coincide. This is derived from a dual form of a theorem of Todorčević. A Lindelöf Σ-space belongs to NFS if and only if its compact subspaces have countable tightness. Hence C_p(X) contains no L-spaces if X is a Lindelöf Σ-space of countable tightness and MA(ω_1) holds, or if X is any Lindelöf Σ-space and PFA holds. Under MA(ω_1), all iterated function spaces of Corson compact spaces are ESLC, and in ZFC the space C_p(X,{0,1}) is ESLC for every zero-dimensional Corson compact space X.
ER -