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On the hyperspace C n ( X ) / C n K ( X )

José G. Anaya, Enrique Castañeda-Alvarado, José A. Martínez-Cortez (2021)

Commentationes Mathematicae Universitatis Carolinae

Let X be a continuum and n a positive integer. Let C n ( X ) be the hyperspace of all nonempty closed subsets of X with at most n components, endowed with the Hausdorff metric. For K compact subset of X , define the hyperspace C n K ( X ) = { A C n ( X ) : K A } . In this paper, we consider the hyperspace C K n ( X ) = C n ( X ) / C n K ( X ) , which can be a tool to study the space C n ( X ) . We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility.

Pasting topological spaces at one point

Ali Rezaei Aliabad (2006)

Czechoslovak Mathematical Journal

Let { X α } α Λ be a family of topological spaces and x α X α , for every α Λ . Suppose X is the quotient space of the disjoint union of X α ’s by identifying x α ’s as one point σ . We try to characterize ideals of C ( X ) according to the same ideals of C ( X α ) ’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let m be an infinite cardinal. (1) Is there any ring R and I an ideal in R such that I is an irreducible intersection of m prime ideals? (2) Is there...

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