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Keywords:
AANR; absolute retract; arc component; arc-like; continuum; decomposable; dendroid; hereditarily unicoherent; retraction; terminal continuum; tree-like
AANR; absolute retract; arc component; arc-like; continuum; decomposable; dendroid; hereditarily unicoherent; retraction; terminal continuum; tree-like
Summary:
Continua that are approximative absolute neighborhood retracts (AANR’s) are characterized as absolute terminal retracts, i.e., retracts of continua in which they are embedded as terminal subcontinua. This implies that any AANR continuum has a dense arc component, and that any ANR continuum is an absolute terminal retract. It is proved that each absolute retract for any of the classes of: tree-like continua, $\lambda $-dendroids, dendroids, arc-like continua and arc-like $\lambda $-dendroids is an approximative absolute retract (so it is an AANR). Consequently, all these continua have the fixed point property, which is a new result for absolute retracts for tree-like continua. Related questions are asked.
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