Fans with the property of Kelley

On the property of Kelley for Hausdorff continua

“…U is an open subset of X, p / ∈ U and p ∈ int(X − U )}. (1) Notice that C({p}, X) ⊂ W. If A ∈ C(X) and A / ∈ C({p}, X), then we choose two disjoint nonempty open subsets U and V of X such that A ⊂ U and p ∈ V . Then A ∈ U ∩ C(X) and p ∈ V ⊂ int(X − U ), and thus (1) follows.…”

“…We show that in metric continua, the definition of maximal limit continuum is equivalent to the definition of Hausdorff maximal limit continuum (Theorem 4.8), and the definition of strong maximal limit continuum is stronger that the definition of Hausdorff strong maximal limit continuum (Proposition 4.13). To end the paper, we show that the equivalences of [1,Theorem 3.11] still hold under these new extensions (Theorem 4.19).…”

“…In 1998, J. J. Charatonik and W. J. Charatonik defined the concepts of maximal limit continuum and strong maximal limit continuum, for metric continua [1,Definitions 3.2 and 3.3]; the authors used those definitions to show several properties of continua having the property of Kelley, and to show that some properties are equivalent to the property of Kelley. In this paper we extend the metric concepts of maximal limit continuum and strong maximal limit continuum, to continua, which we call Hausdorff maximal limit continuum and Hausdorff strong maximal limit continuum, respectively.…”

Property of being semi-Kelley for the cartesian products and hyperspaces

“…A continuum X is said to be semi-Kelley provided that for each subcontinuum K and for every two maximal limit continua M and L in K either M ⊂ L or L ⊂ M . The property of semi-Kelley is a weaker form of the property of Kelley, this property has been introduced and studied in [3] by J.J. Charatonik and W.J. Charatonik (see [2], [1]).…”

“…Thus we have X is Kelley. By[3, Statement 3.17, p. 79], we have that X is semi-Kelley. (2) We consider X × [0, 1] with cylindrical coordinates (r, ϕ, z).…”