## Inverse limits of upper semi-continuous set valued functions.(English)Zbl 1101.54015

Inverse limits are usually defined for sequences $$(X_1,f_1), (X_2,f_2), (X_3,f_3), \cdots$$ such that for each $$i$$, $$X_i$$ is a topological space and $$f_i$$ is a map from $$X_{i+1}$$ to $$X_i$$. In this paper, the authors consider sequences $$(X_1,f_1), (X_2,f_2), (X_3,f_3), \cdots$$ such that for each $$i$$, $$X_i$$ is a compact Hausdorff space and $$f_i$$ is an upper semi-continuous function from $$X_{i+1}$$ to $$2^{X_i}$$. Such a sequence is called an inverse limit sequence and the functions $$f_i$$ are called bonding functions. The inverse limit is defined to be the set of all points $$(x_1, x_2, x_3,\cdots)\in\prod X_i$$ such that $$x_i\in f_i(x_{i+1})$$ for each $$i$$, and is denoted by $$\lim_{\leftarrow}\mathbf{f}$$, where $$\mathbf{f}=f_1, f_2, f_3,\cdots$$. They study properties of inverse limit sequences of compact Hausdorff spaces $$X_i$$ with upper semi-continuous bonding functions $$f_i:X_{i+1}\to 2^{X_i}$$.
Some of the main results are: (1) The limit is non-empty and compact. (2) Assume that for each $$i$$, $$X_i$$ is a Hausdorff continuum and for each $$x\in X_{i+1}$$, $$f_i(x)$$ is connected. Then the limit is a Hausdorff continuum. (3) Suppose $$(X_1,f_1), (X_2,f_2), (X_3,f_3), \cdots$$ and $$(Y_1,g_1), (Y_2,g_2), (Y_3,g_3), \cdots$$ are inverse limit sequences of compact Hausdorff spaces with upper semi-continuous bonding functions. Suppose further that, for each $$i$$, $$\varphi_i:X_i\to Y_i$$ is a map such that $$\varphi_i\circ f_i=g_i\circ\varphi_{i+1}$$. Then the function $$\varphi:\lim_{\leftarrow}\mathbf{f}\to\lim_{\leftarrow}\mathbf{g}$$ defined by $$\varphi(\mathbf{x})=(\varphi_1(x_1), \varphi_2(x_2), \varphi_3(x_3), \cdots)$$ for $$\mathbf{x}=(x_1, x_2, x_3, \cdots)\in\lim_{\leftarrow}\mathbf{f}$$ is continuous. Furthermore, if each $$\varphi_i$$ is injective, so is $$\varphi$$, and if each $$\varphi_i$$ is surjective, so is $$\varphi$$. Some interesting examples in the special case where each $$X_i$$ is the closed unit interval are given.

### MSC:

 54B35 Spectra in general topology 54C60 Set-valued maps in general topology 54B10 Product spaces in general topology