Generalized inverse limits indexed by arbitrary total orders

Steven Clontz

UNC Charlotte

2016 March 11th

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

Abstract

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.

Background and Motivation

Product space: \(X^\omega=[0,1]^{\{0,1,2,\dots\}}\)

Bonding relation: \(f\subseteq_{cl} X^2\)

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

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

Inverse Limit: \(\varprojlim\{X,f,\omega\}\)

• \(\vec x(n)\in f(\vec x(n+1))\)

Assume \(X\) is a Hausdorff continuum

Some properties of \(\varprojlim\{X,f,\omega\}\): [Charatonik and Roe 2014]

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

Totally ordered index: \(\varprojlim\{X,f,L\}\subseteq X^L\)

• \(\vec x(\alpha)\in f(\vec x(\beta))\) 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. \(f=f\circ f\).

Some properties of \(\varprojlim\{X,f,L\}\):

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

Silly example: \(L\) is countable

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

Silly example: the identity relation \(\iota\)

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

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

Counterexample:

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.)

The \(\Gamma\) condition

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

Theorem: Idemptotent, continuum-valued \(f\subseteq I^2\) have \(\Gamma\). [C and Varagona 2015]

Simplest example: \(\gamma\)

The total orders \(\check L\), \(\hat L\)

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 \(\subseteq\).

The LOTS \(\check L\), \(\hat L\)

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.

Theorem: \(\check L\cong \varprojlim\{2,\gamma,L\}\)

Metrizability of \(\varprojlim\{X,f,L\}\)

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.

Characterizing \(\varprojlim\{X,\gamma,L\}\)

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

LOTS \(M\)

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.

Temptation:

\[\varprojlim\{M,\gamma,L\}\cong^?\hat L\times_{lex}M\] 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.

Problems:

• 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.

Solution:

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

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

Future work

• 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 \(\{f_{\alpha,\beta}:\alpha\lt\beta\in L\}\)?

References

• 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.

Thank you!

Slides available at Clontz.org.