A classification of continua and confluent transformation

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1973

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Abstract

We say that a continuous transformation f of a compact metric space X onto a metric space Y is confluent provided that for each continuum K[proper subset]Y every component C of fraised -1 is mapped onto all of K by f. X is a curve is understood to mean that X is a 1-dimensional continuum. A curve X is rational means that X admits a basis of open sets each having countable boundary. The idea of the rim-type of a curve X was introduced by K. Menger and is understood to mean the least ordinal [alpha] such that X admits a basis of open sets each having the property that the [alpha]-th derivative of its boundary is void. The rim—type of a rational curve is a countable ordinal and the rim-type of the arc is one. It is known that every acyclic, rational curve of finite rim-type contains an arc. Answering a question raised by A. Lelek, we have shorn the existence of a chainable, thus acyclic, rational curve of any finite rim-type with the property that each subcurve which is not an arc has rim-type equal to that of the curve. A continuum X belongs to the class "A" is understood to mean that every connected subset of X is path-connected. It is known that if X is in class "A", then X is rational. We have shown that the confluent image of a continuum X belonging to class "A" is again in class "A". A curve X is regular means that X admits a basis of open sets each having finite boundary. A continuous transformation f of a compact metric space X onto a metric space Y is weakly confluent provided that, if K is a continuum in Y, then there exists a continuum C in X such that f(C) = K...

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