Countable connected spaces and bunches of arcs in

We investigate the images (also called quotients) of countable connected bunches of arcs in R 3 , obtained by shrinking the arcs to points (see Section 2 for definitions of new terms). First, we give an intrinsic description of such images among T 1 -spaces: they are precisely countable and weakly first countable spaces. Moreover, an image is first countable if and only if it can be represented as a quotient of another bunch with its projection hereditarily quotient (Theorem 2.7). Applying this result we see, for instance, that two classical countable connected T 2 -spaces—the Bing space [R.H. Bing, A connected countable Hausdorff space, Proc. Amer. Math. Soc. 4 (1953) 474], and the Roy space [P. Roy, A countable connected Urysohn space with a dispersion point, Duke Math. J. 33 (1966) 331–333]—belong to such images. However, in these cases, we can show even more: each of the examples is a quotient, with hereditarily quotient projection, of a countable bunch of free segments (Examples 2.12 and 2.15). Next, we construct an example of a countable connected planar bunch of segments whose quotients are not first countable (Theorem 2.9). We also construct a collection of power c of countable connected Hausdorff spaces (with some extra properties). As a corollary we get that there exists a collection of power c of countable connected bunches of arcs in R 3 no two of which are homeomorphic (Theorem 3.1). We end this article with some open problems.