# The Point of Continuity Property: Descriptive Complexity and Ordinal Index

Bossard, Benoit; López, Ginés

Serdica Mathematical Journal (1998)

- Volume: 24, Issue: 2, page 199-214
- ISSN: 1310-6600

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topBossard, Benoit, and López, Ginés. "The Point of Continuity Property: Descriptive Complexity and Ordinal Index." Serdica Mathematical Journal 24.2 (1998): 199-214. <http://eudml.org/doc/11590>.

@article{Bossard1998,

abstract = {∗ Supported by D.G.I.C.Y.T. Project No. PB93-1142Let X be a separable Banach space without the Point of
Continuity Property. When the set of closed subsets of its closed unit ball
is equipped with the standard Effros-Borel structure, the set of those which
have the Point of Continuity Property is non-Borel. We also prove that,
for any separable Banach space X, the oscillation rank of the identity on
X (an ordinal index which quantifies the Point of Continuity Property) is
determined by the subspaces of X with a finite-dimensional decomposition.
If X does not contain l1 , subspaces with basis suffice. If X ∗ is separable,
one can even restrict to subspaces with shrinking basis.},

author = {Bossard, Benoit, López, Ginés},

journal = {Serdica Mathematical Journal},

keywords = {Point of Continuity Property; Borel Set; Ordinal Index; point of continuity property; Borel set; ordinal index; Effros-Borel structure; oscillation rank; finite-dimensional decomposition; shrinking basis},

language = {eng},

number = {2},

pages = {199-214},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {The Point of Continuity Property: Descriptive Complexity and Ordinal Index},

url = {http://eudml.org/doc/11590},

volume = {24},

year = {1998},

}

TY - JOUR

AU - Bossard, Benoit

AU - López, Ginés

TI - The Point of Continuity Property: Descriptive Complexity and Ordinal Index

JO - Serdica Mathematical Journal

PY - 1998

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 24

IS - 2

SP - 199

EP - 214

AB - ∗ Supported by D.G.I.C.Y.T. Project No. PB93-1142Let X be a separable Banach space without the Point of
Continuity Property. When the set of closed subsets of its closed unit ball
is equipped with the standard Effros-Borel structure, the set of those which
have the Point of Continuity Property is non-Borel. We also prove that,
for any separable Banach space X, the oscillation rank of the identity on
X (an ordinal index which quantifies the Point of Continuity Property) is
determined by the subspaces of X with a finite-dimensional decomposition.
If X does not contain l1 , subspaces with basis suffice. If X ∗ is separable,
one can even restrict to subspaces with shrinking basis.

LA - eng

KW - Point of Continuity Property; Borel Set; Ordinal Index; point of continuity property; Borel set; ordinal index; Effros-Borel structure; oscillation rank; finite-dimensional decomposition; shrinking basis

UR - http://eudml.org/doc/11590

ER -

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