General properties of pseudo-contractibility

Ri-sets, pseudo-contractibility and weak contractibility on hyperspaces of continua

“…The last proposition gives a partial answer to [3,Question 64]. The main problem in this section is to prove that the notions of weak contractibility (pseudo-contractibility) and contractibility coincide in the hyperspaces 2 X , C(X), C ∞ (X) and C n (X) for any n ∈ N.…”

“…On the other hand, we know that if n > 1, F 2n+1 (S 1 ) is homotopically equivalent to S 2n+1 (see [6,Theorem 4.1]) and F 2n (S 1 ) is homotopically equivalent to S 2n−1 (see [6,Theorem 4.2]). Since S n is ANR for all n ∈ N and S n is not contractible, then by [3,Corollary 46], S n is not pseudo-contractible. Therefore, by [3, Corollary 39], F n (S 1 ) is not pseudocontractible for all n ∈ N.…”

“…C(X) is contractible, because X is a Kelley's continuum (see [8,Theorem 9.4, p. 129]). By [4], F 2 (X) is not unicoherent and by [3,Corollary 56], it is not pseudo-contractible.…”

“…It is well known by [15,Theorem 3.4] that if X is a proper circle-like continuum, then F 2 (X) does not have trivial shape. Therefore, if X is a proper circle-like continuum, then by [3,Theorem 48], F 2 (X) is not pseudocontractible. On the other hand, by [15,Theorem 3.3] if there exists an essential map from a continuum X onto S 1 , then there exists an essential map from F 2 (X) onto S 1 , in other words, F 2 (X) does not have trivial shape and therefore, by [3, Theorem 52], F 2 (X) is not pseudo-contractible.…”

“…In particular the pseudo-arc is another example of a non pseudo-contractible continuum. In [3] there is a general study about pseudo-homotopies and pseudo-contractibility. The interested reader is referred to [2,3,7,9,11] and [19].…”