# Inverse limits with upper semi-continuous bonding functions problems

These problems come from this pdf by W.T. Ingram.

Problem 2.2 (ILWUSCBF): Suppose $f$ is a sequence of upper semi-continuous functions on a Hausdorff continua. Find necessary and sufficient conditions (preferably on the bonding functions) such that $\varprojlim f$ is connected.

Problem 2.3 (ILWUSCBF): Suppose $X$ is a compact metric space and $f \colon X \rightarrow 2^X$ is an upper semi-continuous function. Find sufficient conditions on $f$ such that $\varprojlim f$ is connected.

Problem 3.2 (ILWUSCBF): Find conditions on the bonding functions that ensure that closed subsets of the inverse limit are the inverse limit of their projections.

Problem 3.3 (ILWUSCBF): Find conditions on the bonding functions that ensure that an inverse limit has the full projection property.

Problem 4.1 (ILWUSCBF): Find sufficient conditions on the bonding functions so that the inverse limit $\varprojlim f$ is indecomposable.

Problem 5.1 (ILWUSCBF): If $X$ is a sequence of compact Hausdorff spaces and $\vec{f}$ is a sequence of upper semi-continuous functions such that $f_i \colon X_{i+1} \rightarrow 2^{X_i}$ (or $f_i \colon X_{i+1} \rightarrow C(X_i)$) for each positive integer $i$, and $n_1,n_2,n_3,\ldots$ is an increasing sequence of positive integers, find sufficient conditions on the bonding functions such that if $g_i=f_{n_i n_{i+1}}$ for each $i$, then $\varprojlim \vec{f}$ and $\varprojlim \vec{g}$ are homeomorphic.

Problem 5.2 (ILWUSCBF): If $f \colon X \rightarrow 2^X$ is an upper semi-continuous function on a compact Hausdorff space $X$ and $n$ is a positive integer greater than $1$, find sufficient conditions such that $\varprojlim \vec{f}$ is homeomorphic to $\varprojlim \vec{f}^n$.
Problem 6.1 (ILWUSCBF): For a given compact Hausdorff space $X$, which compact sets are not homeomorphic to inverse limits with a single upper semi-continuous function $f \colon X \rightarrow 2^X$ ($f \colon X \rightarrow C(X)$)?
Problem 7.1 (ILWUSCBF): Find conditions on the bonding functions that ensure that $$\{\pi_i^{-1}(O) \colon i \mathrm{ \hspace{2pt} is \hspace{2pt} a \hspace{2pt} positive \hspace{2pt} integer \hspace{2pt} and \hspace{2pt}} O \mathrm{\hspace{2pt} is \hspace{2pt} open \hspace{2pt} in \hspace{2pt}} X_i\}$$ is a basis for the topology of the inverse limit with upper semi-continuous bonding functions that are not mappings.