Countable connected spaces and bunches of arcs in R3

We investigate the images (also called quotients) of countable connected bunches of arcs in R3, 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 T1-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 T2-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 R3 no two of which are homeomorphic (Theorem 3.1). We end this article with some open problems.