AANR spaces and absolute retracts for tree-like continua

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

Confluent Mappings and Arc Kelley Continua

A Kelley continuum X, also called a continuum with the property of Kelley, such that, for each p X, each subcontinuum K containing p is approximated by arc-wise connected continua containing p, is called an arc Kelley continuum. A continuum homeomorphic to the inverse limit of locally connected continua with confluent bonding maps is said to be confluently LC-representable. The main subject of the paper is a study of deep connections between the arc Kelley continua and confluent mappings. It is shown that if a continuum X admits, for each ε \u3e 0, a confluent ε-mapping onto a(n) (arc) Kelley continuum, then X itself is a(n) (arc) Kelley continuum. In particular each confluently LC-representable continuum is arc Kelley. It is also proved that if continua X and Y are confluently LC-representable, then also are their product X x Y and the hyperspaces 2^x and C(X)

Hereditarily Unicoherent Continua and Their Absolute Retracts

We investigate absolute retracts for classes of hereditarily unicoherent continua, tree-like continua, λ- dendroids, dendroids and some other related ones. The main results are: (1) the inverse limits of trees with confluent bonding mappings are absolute retracts of hereditarily unicoherent continua; (2) each tree-like continuum is embeddable in a special way in a tree-like absolute retract for the class of hereditarily unicoherent continua; (3) a dendroid is an absolute retract for hereditarily unicoherent continua if and only if it can be embedded as a retract into the Mohler-Nikiel universal smooth dendroid

The lifting property for classes of mappings

The lifting property of continua for classes of mappings is defined. It is shown that the property is preserved under the inverse limit operation. The results, when applied to the class of confluent mappings, exhibit conditions under which the induced mapping between hyperspaces is confluent. This generalizes previous results in this topic

Decompositions and the fixed point property for multifunctions

Relations between the fixed point properties for some classes of multifunctions of a compact Hausdorff space X, of a decomposition space X/D, where D is an upper semi-continuous decomposition of X, and of the members of D are studied. Results are applied to some special decompositions of metric continua

Many continuous functions have many proper local extrema

AbstractGiven a topological space X, let M(X) (resp. m(X)) denote the set of all continuous real functions on X whose set of proper local maximum (resp. minimum) points is dense in X. We identify some classes of spaces X for which M(X) is a dense subset of C(X) endowed with the majorant topology. In particular, M(X) ∩ m(X) is dense in C(X) with the majorant topology whenever X has a σ-discrete π-base and a dense subset whose points are Gδ-sets.Also we show that M(X) ∩ m(X) is residual in C(X) endowed with the topology of uniform convergence, provided that X has a σ-discrete π-base consisting of completely metrizable subspaces. This is true, in particular, for all completely metrizable spaces

Atomoicity of mappings

A mapping f:X→Y between continua X and Y is said to be atomic at a subcontinuumK of the domain X provided that f(K) is nondegenerate and K=f−1(f(K)). The set of subcontinua at which a given mapping is atomic, considered as a subspace of the hyperspace of all subcontinua of X, is studied. The introduced concept is applied to get new characterizations of atomic and monotone mappings. Some related questions are asked

Smoothness and the property of Kelley

summary:Interrelations between smoothness of a continuum at a point, pointwise smoothness, the property of Kelley at a point and local connectedness are studied in the paper