Aposyndesis and coherence of continua under refinable maps

Some theorems on colocally connected continua

“…It is known that a continuum X is aposydetic if and only if X is semilocally connected (see [10]). Hence, Theorem 2.3 implies the following result proven by Hosokawa [8] and vice versa. If (X, d) is a space, x ∈ X and ε > 0, then we denote the set…”

“…In [8], Hosokawa proved that each refinable map defined on a continuum preserves aposyndesis. Also, in [14] Petrus proved that aposyndesis is a Whitney property and is not a Whitney reversible property.…”

“…([8, Corollary (p.368)]) Let X and Y be continua and let f : X → Y be a refinable map. If X is aposyndetic, then so is Y .…”

Uniformly refinable maps

“…The following theorem extends [14,Theorem 4.1.32] to uniformly atomically refinable maps and gives a partial positive answer to [14,Question 8.1.18]. The following theorem is an extension to closed sets of [7,Theorem 1 ]. Its proof is similar to the one given for Theorem 4.1; we include the details to see the "duality" of T and K. Theorem 4.5.…”