On the n-fold symmetric product suspensions of a continuum

Abstract

In 1979 Sam B. Nadler, Jr. defined the hyperspace suspension of a continuum. We define the n-fold symmetric product suspensions of a continuum using n-fold symmetric products. We study some properties of this hyperspace: unicoherence, local connectedness, arcwise connectedness.

More on induced maps on n-fold symmetric product suspensions

“…Also, if Y is locally connected and n is greater than or equal to 3, then (2) implies (1), (3) implies (2) and (3) implies (1). Moreover, if n is greater than or equal to 3, (1) does not imply (2) and (1) does not imply (3).…”

“…• strongly freely decomposable if whenever A and B are proper subcontinua of Y such that Y = A ∪ B, we obtain that f −1 (A) and f −1 (B) are connected 1 .…”

“…Then the n-fold symmetric product suspension ( [1]) of a continuum X, denoted by SF n (X), is the quotient space F n (X)/F 1 (X), with the quotient topology. The quotient map is denoted by q n X : F n (X)→ →SF n (X) and we denote q n X (F 1 (X)) by F n X .…”

Dynamic properties of the dynamical system SFnm(X), SFnm(f))

“…Subsequently, the space C n (X)/F 1 (X) was studied [26]. In 2010 [4], the first named author of this paper defined the space F n (X)/F 1 (X) which is denoted by SF n (X) and is called the n-fold symmetric product suspension of the continuum X. Some topological properties of SF n (X) are studied in [4,6].…”

Induced maps on n-fold symmetric product suspensions

“…In 2004 S. Macías, defined the n-fold hyperspace suspension of a continuum [2]. For a continuum X and n 2, in 2009, we define the n-fold symmetric product suspensions of X [3], denoted by SF n (X), as the quotient space F n (X)/F 1 (X), where F n (X) is the hyperspace of nonempty subsets of X with at most n points. Given a map f : X → Y between continua and an integer n 2, we let F n ( f ) : F n (X) → F n (Y ) and SF n ( f ) : SF n (X) → SF n (Y ) denote the corresponding induced maps.…”