2016 March 11th

Presentation for the
**
Spring Topology and Dynamics Conference 2016
Special Session on Continuum Theory
**
at
**
Baylor University
**

Call an idempotent uppersemicontinuous continuum-valued surjective relation on \(X^2\) a CV-relation. The presenter and S. Varagona showed that an inverse limit of a linearly ordered compactum indexed by an ordinal and bonded with a single CV-relation is metrizable if and only if the ordinal is countable. This result may be generalized to any totally ordered index.To demonstrate this, the presenter will give a simple characterization for the inverse limit bonded by the simple CV-relation \(\gamma\) in terms of the lexicographic product of the factor space and linearly ordered index.

A.K.A. u.s.c. bonding map \(f:X\to C(X)\)

- \(f\) is surjective.
- \(f\) is full.

- \(\vec x(n)\in f(\vec x(n+1))\)

Assume \(X\) is a Hausdorff continuum

- Nonempty
- Metrizable (if \(X\) is)
- Compact
- Connected (if \(f\) continuum-valued)

- \(\vec x(\alpha)\in f(\vec x(\beta))\) for all \(\alpha\lt\beta\) in \(L\)
- Assume \(f\) is idempotent: \(f(x)=\{z:\exists y\in f(x)\text{ such that }z\in f(y)\}\), i.e. \(f=f\circ f\).

- Nonempty
- Compact
- Connected (if \(f\) continuum-valued)
- Hausdorff
~~Metrizable~~(?)

\(\varprojlim\{X,f,L\}\subseteq X^L\) is metrizable.

\(\varprojlim\{X,\iota,L\}\cong X\) is metrizable

We now assume \(f\not=\iota\).

Varagona first showed that \(\varprojlim\{I,\gamma,\omega_1\}\) is the closed long ray of length \(\omega_1\), so not metrizable (or even Corson compact, \(W\), Frechet-Urysohn, first-countable, etc.)

There exist \(x,y\in I\) such that \(\langle x,x\rangle,\langle x,y\rangle,\langle y,y\rangle\) are all in \(f\).

For any total order \(L\), we may define \(\check L=\{A\subseteq L:a\in L,b\lt a\Rightarrow b\in L\}\) and \(\hat L=\{A\in\check L:A\text{ is closed}\}\), which are both totally ordered by \(\subseteq\).

Give \(\check L\) its usual order topology generated by the sets \((A,B)=\{C\in\check L:A\subsetneq C\subsetneq B\}\). (Sim. for \(\hat L\).)

Note \(\check L,\hat L\) are always compact spaces.

If \(f\) has \(\Gamma\), then \(\varprojlim\{X,f,L\}\supseteq\varprojlim\{2,\gamma,L\}=\check L\)

FACT: \(\check L\) is metrizable iff \(\check L\) is Corson compact iff \(\check L\) is second-countable iff \(L\) is countable

Therefore, \(\varprojlim\{X,f,L\}\) cannot be metrizable or even Corson compact unless \(f\) lacks \(\Gamma\) or \(L\) is countable.

\(\check L\) contains the lexicographic product \(L\times_{lex} 2\), adding new points for leftward sets without a supremum.

Suppose \(M\) is a (compact) LOTS with minimum \(0\) and maximum \(1\).

A point in \(\varprojlim\{M,\gamma,L\}\) is a thread which is valued \(1\) on some closed leftward set \(A\in\hat L\), except it may have any value of \(M\) on its supremum (if it exists), and \(0\) otherwise.

\[\varprojlim\{M,\gamma,L\}\cong^?\hat L\times_{lex}M\] where \(\langle A,m\rangle\) corresponds to the thread valued \(m\) on \(\sup A\), valued \(1\) for points in \(A\setminus\{\sup A\}\), and valued \(0\) otherwise.

- If \(l,l+1\in L\) are successors, we cannot have separate points for the single thread where \(\vec x(l)=1\) and \(\vec x(l+1)=0\).
- If \(A\in\hat L\) has no supremum, its corresponding thread is forced to be \(1\) on \(A\) and \(0\) otherwise, so should correspond to a single point.

\(\varprojlim\{M,\gamma,L\}\cong(\hat L\times_{lex}M )/ \sim\)

- \( \langle(\leftarrow,l),1\rangle \sim \langle(\leftarrow,l],0\rangle \)
- When \(A\in\hat L\) has no supremum or \(A=\emptyset\), \( \langle A,m\rangle \sim \langle A,m'\rangle \)

- It can be shown that \( \varprojlim\{M,\nu,L\} \cong (\check L\times M)/\sim \) where \(\langle\emptyset,m\rangle\sim\langle A,0\rangle\). Any other simple computations?
- Can it be shown that any idempotent \(f\) satisfies condition \(\Gamma\)?
- Can similar techniques be used for a family of bonding relations \(\{f_{\alpha,\beta}:\alpha\lt\beta\in L\}\)?

- Wlodzimierz J. Charatonik and Robert P. Roe, On Mahavier Products, Topology and its Applications, 166, (2014), 92-97.
- Steven Clontz, Characterizations of Generalized Inverse Limits Indexed by Total Orders, in preparation
- Steven Clontz and Scott Varagona, Destruction of Metrizability in Generalized Inverse Limits, Topology Proc. 48 (2016), 289-297.
- Sina Greenwood and Judy Kennedy, Connected generalized inverse limits, Topology and its Applications, 159 (2012), no. 1, 57-68.
- W. T. Ingram and William S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston Journal of Mathematics, vol. 32 (2006) no. 1, 119-130.
- Van Nall, Connected inverse limits with a set-valued function, Topology Proc. 40 (2012), 167-177.
- Scott Varagona, Generalized Inverse Limits Indexed by Totally Ordered Sets, http://arxiv.org/abs/1511.00266
- Patrick Vernon, Inverse limits of set-valued functions indexed by the integers, Topology Applications 171 (2014), 35-40.

Slides available at Clontz.org.