On span and inverse limits
Kazuhiro Kawamura · Tsukuba Journal of Mathematics · 1988
A compact metric space is called a compactum and a connected compactum is called a continuum.All maps in this paper are continuous.Let $f:X\rightarrow Y$ be a map between continua.Ingram [2] and Lelek [11] defined the span, semispan, surjective span, and surjective semispan of $f$ by the following formulaS (the mapwhere the condition $\tau$ ) is:The span of a continuum $X$ is defined by $\sigma(id_{X})$ .The other cases are similar.In the same way, we can define the symmetric span of $f$ by the formula $s(f)=\sup\{c\geqq 0|_{andd(f(x),f(y))\geqq cforeach(x,y)\in Z^{ch}}^{thereexistsacontinuumZ\subset X\times X}Zissymmetric(i.e.(x,y)\in Ziff(y^{su}x)\in^{th}Z^{a})^{t}\}$ .It is a mapping version of symmetric span of a continuum due to J. F. Davis [1].Let $X=\lim_{\leftarrow}(X_{n}, p_{nn+1})$ be a continuum, where $p_{nn+1}$ : $X_{n+1}\rightarrow X_{n}$ .Ingram [ 2] and [4] showed that $\sigma(X)=0$ if and only if there exists a cofinal subsequence $(n_{i})_{i\geq 1}$ such that $\lim_{j}\sigma(p_{n_{i}n_{j}})=0$ for each $i\geqq 1$ .In section 2 of this paper, we will prove a mapping version of this theorem.H. Cook proved essentially that the symmetric span of the dyadic solenoid is zero ([1], p. 134), while its span is positive.The author wishes to thank to the referee for pointing out this fact.In section 3, we generalize this to the poly-adic solenoid.Let $f$ and $g:X\rightarrow Y$ be maps.$d(f, g)$ denotes $\sup\{d(f(x), g(x))|x\in X\}$ .