Continuous Images of Arcs and Inverse Limit Methods by Jacek Nikiel

By Jacek Nikiel

Non-stop photographs of ordered continua were studied intensively considering that 1960, whilst S. Mardsic confirmed that the classical Hahn-Mazukiewicz theorem doesn't generalize within the 'natural' method to the non metric case. In 1986, Nikiel characterised acyclic pictures of arcs as continua which might be approximated from inside of via a chain of well-placed subsets which he referred to as T-sets. That characterization has been used to reply to a number of exceptional questions within the zone. during this publication, Nikiel, Tymchatyn, and Tuncali examine photographs of arcs utilizing T-set approximations and inverse limits with monotone bonding maps. a couple of very important theorems on Peano continua are prolonged to photographs of arcs. the various effects awarded listed below are new even within the metric case.

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Clearly, each /„ is a monotone surjection. Let n be a positive integer. Observe that each non-degenerate cyclic element of Xn is Yn for some Y G Lx- Hence, each cyclic element of Xn is rim-finite. Therefore Xn is rim-finite. 4 gives in fact a characterization of 1dimensional spaces which are continuous images of arcs. 5. Corollary. Let X be a continuum. Then the following conditions are equiva- lent: (a) X is a continuous image of an arc and dim X — \, (b) X is homeomorphic to lim inv S for some inverse sequence S = (Xn,fn) finite continua Xn with monotone bonding surjections fn.

For each of M(i) and -/V(i), exactly one of the following three possibilities holds: it is a member of £K(i)> o r it i s a union of two distinct members of £x(i) (which meet at x or y, respectively), or it is degenerate. , i -f 1 = m = n), then there is a positive integer £(i) such that x £ Ex and t / 6 ^ for A*, Ey G £J (i) such that Ex f) Ey = Hi and A^ fl bd(/v (i)) = 0 = Ay fl bd(A'(f)). ')) = 0. , z + 1 < m < n), then there is £(i) such that K(i + 1) fl E = 0 = L(i -f 1) H A for each A* G £J ( 0 .

By for i — 1 , . . , n. Since ^4,- is perfect, A* contains a perfect subset. If yi G A* for i = 1, .. , n, then { y i , . , yn] separates x from y. T h u s , X is totally regular. 8. Lemma. Fori G {1,2}, let Li be a (metric) let Wi : L\ x Lo —+ L; denote the projection.

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