There are theorems in which some classes of topological spaces are
characterized by means of properties of mappings of these spaces into a single
space. For example, it is well known that a compactum X
is at most n-dimensional if and only if no mapping of X
irto an (n + l)-cube has a stable value [5, Theorems
VI. 1-2, pp. 75-77]. Also, a curve X is tree-like if
and only if no mapping of X into a figure eight is
homotopically essential [1, Theorem 1, pp. 74-75;
8, p. 91]. By a curve we mean
any at most 1-dimensional continuum; a continuum is a
connected compactum; a compactum is a compact metric
space, and a mapping is a continuous function. The aim
of the present paper is to prove another theorem of this type. We distinguish a
class of curves and show that it is characterized by imposing the condition that
no weakly confluent mapping [13] can transform the given
curve onto a simple triod (see 2.4). A related result is applied to a generalized
branch-point covering theorem (see 3.2). In addition, two results are obtained in
which we establish some characterizations of weakly confluent images and preimages
of the product of the Cantor set and an arc (see 1.1 and 2.2). Continua that are
such images turn out to be identical with regular curves (see 1.3).