Continuous Images of Arcs and Inverse Limit Methods by Jacek Nikiel

By Jacek Nikiel

Non-stop photographs of ordered continua were studied intensively due to the fact that 1960, while S. Mardsic confirmed that the classical Hahn-Mazukiewicz theorem doesn't generalize within the 'natural' technique to the non metric case. In 1986, Nikiel characterised acyclic photos of arcs as continua that are approximated from inside of by way of a chain of well-placed subsets which he referred to as T-sets. That characterization has been used to reply to a bunch of exceptional questions within the region. during this e-book, Nikiel, Tymchatyn, and Tuncali research photos of arcs utilizing T-set approximations and inverse limits with monotone bonding maps. a few very important theorems on Peano continua are prolonged to pictures of arcs. many of the effects provided listed below are new even within the metric case.

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Example text

T h e proof of Claim 7 is complete. Let m be a positive integer. Let Qm denote the decomposition of X into members of Tm and points. Since Tm is a null-family, Qm is upper semi-continuous. Let Ym denote the quotient space Ym = X/n and pm : X —• Ym be the quotient m a p . T h e n Ym is a locally connected continuum and pm is monotone. By Claim 7, for each positive integer m, there exists th e unique function qm : Y m +i —*• Ym such t h a t p m = qm o p m + 1 . Obviously, g m is a continuous surjection.

X,y) from x to y. We may suppose withou t loss of generality t h a t z G cl({u; G E : w < z}). Let {z6 :6<6} be an increasing transfinite sequence in E which converges to z. Let f3 > Qfo. For each 8 < 6, x f<*op( ) f rom z an faop( ) follows t h a t Ls / ~ p(zs) l fa^i26) i ls a n e t h a t Ls f~lp(zs) d from / ^ ( y ) . 7. Proof. — \zp\ xao Since faop is continuous and lim 2^ = z it ls a point. Since Xp is locally connected it follows t h a t Hence, {z*} = l i m / J 1 [zyj separates z\ f~*(x) in X .

G £". Then E ^ G ^ Ef. By Claim 9, there are finite collections 7i C Tm - {G,E}, ft' C ^ m - { G , ^ } , n0 C Tm - {G,F} and fti C Tm — {G, F} such that G C\ L and E C\ L are in distinct components of L — (J ft, G f l M and F ' D A/ are in distinct components of M - {JH*, G fl L and F C\ L are in distinct components of L — IJfto provided F D L ^ 0, and G fl M and F fl M are in distinct components of M - (J fti provided F fl M ^ 0 (we let ft0 = 0 if F n L = 0 and fti = 0 if FDAf = 0; note that at most one of the inequalities FC\L / 0 and FflAf ^ 0 can hold).

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