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Inverse limits with upper semi-continuous bonding functions whose inverse functions are continuous. (English) Zbl 1442.54027

In the present paper the authors study generalized inverse limits with upper semi-continuous bonding functions whose inverse functions are continuous. In the first part of the paper, under the condition that each factor space is an arbitrary compactum, one can find a sufficient condition for inverse sequences to have the full projection property. More precisely, it is proven that if the bonding function \(f_i\) is surjective, \(f_i^{-1}\) is continuous, and the set \(A'(f_i) = \{(x,y) \in G(f_i) \ | \ f_i(x) \text{ is a singleton}\}\) is dense in the graph \(G(f_i)\) for each positive integer \(i\), then the corresponding inverse sequence has the full projection property.
In the second part of the paper, the authors study the properties of the product of inverse limits (i.e.if \(X_1\), \(X_2\) \(Y_2\), and \(Y_2\), are compact metric spaces and \(f: X_1 \to 2^{Y_1}\), \(g: X_2 \to 2^{Y_2}\) upper semi-continuous functions, then the upper semi-continuous function \(f \times g : X_1 \times X_2 \to 2^{Y_1 \times Y_2}\) is defined by \((f \times g)(x_1, x_2) = f(x_1)\times g(x_2)\); by analogy the product of generalized inverse limits is defined). Based on the result mentioned above, the authors give a sufficient condition for the product of inverse sequences to have the full projection property. In the same section, we can find two inverse sequences having the continuum full projection property, although their product does not. Also, they pose the following open problems:
If \(\{X_i,f_i\}_{i=1}^{\infty}\) and \(\{Y_i,g_i\}_{i=1}^{\infty}\) have the full projection property, then does \(\{X_i \times Y_i,f_i \times g_i\}_{i=1}^{\infty}\) have the full projection property? If \(\{X_i \times Y_i,f_i \times g_i\}_{i=1}^{\infty}\) has the (continuum) full projection property, do then \(\{X_i,f_i\}_{i=1}^{\infty}\) and \(\{Y_i,g_i\}_{i=1}^{\infty}\) have the (continuum) full projection property?
In the remaining part of the paper, we can find some applications of the introduced theory related to open mappings, indecomposability, etc. In particular, the authors discuss relations between the proven theorems and results of I. Banič et al. [Mediterr. J. Math. 15, No. 4, Paper No. 167, 1–21 (2018; Zbl 1404.54017)]. Among other things, they give a counterexample to an open problem posed by Banič et al. Also, the authors state some open problems.

MSC:

54F17 Inverse limits of set-valued functions
54F15 Continua and generalizations
54C60 Set-valued maps in general topology

Citations:

Zbl 1404.54017
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References:

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