## Metrizability in Generalized Inverse Limits

#### Steven Clontz

2016 March 3rd

Presentation for the University of South Alabama

### Abstract [PDF]

For the metric arc I=[0,1] and continuum-valued bonding relation f closed in I^2, the inverse limit lim{I,f,ω} is the subspace of the countable power I^ω containing sequences x satisfying x(n)∈f(x(n+1)). A recent trend in continuum theory is to generalize this notion to lim{I,f,L}, where L is an arbitrary linear order. When L=ω, the inverse limit is a subspace of the metrizable space I^ω; however, we will show that when L is uncountable, the inverse limit cannot be metrizable unless f is trivial. Furthermore, when L is an uncountable well order, it will be shown that the inverse limit is not even Corson compact.

## Background and Motivation

#### Bonding map/relation: $$f\subseteq_{cl} I^2$$

• $$f(x)=[a_x,b_x]$$ for all $$x\in[0,1]$$. (so $$f$$ is connected)
• $$f$$ is surjective.
• If $$a_x=b_x$$ for all $$x\in[0,1]$$, then $$f$$ corresponds to a continuous function.

#### Inverse Limit: $$\varprojlim\{I,f,\omega\}$$

• $$\vec x(n)\in f(\vec x(n+1))$$

• Nonempty
• Metrizable
• Compact
• Connected

#### Totally ordered index: $$\varprojlim\{I,f,L\}\subseteq I^L$$

• $$\vec x(\alpha)\in f(\vec x(\beta))$$ 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 $$\varprojlim\{I,f,L\}$$:

• Nonempty
• Compact
• Connected
• Hausdorff
• Metrizable (?)

#### Silly example: $$L$$ is countable

$$\varprojlim\{I,f,L\}\subseteq I^L$$ is metrizable.

#### Silly example: the identity relation $$\iota$$

$$\varprojlim\{I,\iota,L\}\cong I$$ is metrizable

We now assume $$f\not=\iota$$.

#### Counterexample: $$f=\gamma$$, $$L=\omega_1$$

$$\varprojlim\{I,\gamma,\omega_1\}$$ is the closed long ray of length $$\omega_1$$, so not metrizable (or even Corson compact, $$W$$, Frechet-Urysohn, 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 $$f$$ have $$\Gamma$$

#### If $$\iota\subsetneq f$$...

$$\Gamma$$ trivially holds

#### Can $$f$$ miss $$\iota$$ completely?

Contradicts connectedness of $$f$$.

#### Can $$f$$ only hit one point on $$\iota$$?

Contradicts surjectivity and idempotence of $$f$$.

### So $$f\cap\iota$$ hits two points.

#### Two cases to handle...

1. $$f\cap\iota$$ is connected
2. $$f\cap\iota$$ is disconnected

#### Case 1: $$f\cap\iota$$ is connected

Contradicts surjectivity and idempotence of $$f$$.

#### Case 2: $$f\cap\iota$$ is disconnected

Contradicts idempotence of $$f$$.

## Applying condition $$\Gamma$$

Now that $$\Gamma$$ has been verified for our bonding relations, we may restrict our attention to the two-point discrete space $$2=\{0,1\}$$ and investigate $$\varprojlim\{2,\gamma,L\}\subseteq\varprojlim\{X,f,L\}$$.

### The total order $$\check L$$

For any total order $$L$$, we may define $$\check L=\{A\subseteq L:a\in L,b\lt a\Rightarrow b\in L\}$$ which is totally ordered by $$\subseteq$$.

### The LOTS $$\check L$$

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

Note $$\check L$$ is always a compact space.

### $$\check L\cong\varprojlim\{2,\gamma,L\}$$

Note that for $$\vec x\in\varprojlim\{2,\gamma,L\}$$, a value of $$0$$ may stay $$0$$ or change to $$1$$ as we look to the left.

But, once the value changes to $$1$$, it's forced to stay there.

It follows that each $$\vec x$$ is exactly the characteristic function $$\phi_A$$ for some $$A\in \check L$$; that is, $\vec x(l)=\phi_A(l)=\begin{cases} 1 & \text{if } l\in A \\ 0 & \text{if } l\not\in A \end{cases}$

Note further that in the topology on $$\varprojlim\{2,\gamma,L\}\subseteq 2^L$$, a basic open set may fix $$\vec x(a)=1$$ and $$\vec x(b)=0$$ for some $$a\lt b \in L$$.

This is exactly the basic open set $$((\leftarrow,a),(\leftarrow,b])$$ in $$\check L$$.

Thus the map $$A\mapsto\phi_A$$ is a homeomorphism from $$\check L$$ to $$\varprojlim\{2,\gamma,L\}$$.

## Exploiting $$\check L\subseteq\varprojlim\{I,f,L\}$$

Since we've found a copy of the compact space $$\check L$$, we can break metrizability and other properties by showing that they cannot hold in $$\check L$$.

### Ordinals

Consider the totally ordered ordinal spaces such as $$0=\emptyset$$, $$3=\{0,1,2\}$$, $$\omega=\{0,1,2,\dots\}$$, $$\omega+2=\{0,1,\dots,\omega,\omega+1\}$$ ordered by $$\subseteq$$.

Since $$\alpha+1=\alpha\cup\{\alpha\}$$, it's not hard to see that $$\check\alpha=\alpha+1$$ for every ordinal, including the first uncountable ordinal $$\omega_1$$.

### $$W$$-spaces

It's easy to show that every countable $$\alpha+1$$ is metrizable.

But in fact, even the first uncountable successor $$\omega_1+1$$ is not metrizable, or even a $$W$$-space.

### $$Gru_{O,P}(X,x)$$

Player $$\mathcal O$$ wins if after $$\omega$$ arounds, $$\lim_{n\to\infty}x_n=x$$, and $$\mathcal P$$ wins otherwise. $$x$$ is a $$W$$-point if $$\mathcal O$$ has an unbeatable strategy at that point.

In a metrizable space, it's obvious that every point is $$W$$ since $$\mathcal O$$ can simply choose radii converging to $$0$$. Since every point is $$W$$, it's a $$W$$ space.

### $$Gru_{O,P}(\check\omega_1,\omega_1)$$

Because the game only lasts countably many rounds, $$\mathcal P$$'s' countable sequence must have an upper bound $$\beta$$ in $$\omega_1$$. Therefore $$(\beta,\rightarrow)$$ is a neighborhood of $$\omega_1$$ missing every point in $$\mathcal P$$'s sequence.

### Nonmetrizability of $$\check L$$, $$\varprojlim\{I,f,L\}$$

We've shown that $$\check\alpha$$ is metrizable (or even Corson compact, $$W$$, Fréchet–Urysohn, etc. etc.) if and only if $$\alpha$$ is countable.

More generally, it can be shown that $$\check L$$ is metrizable if and only if $$L$$ is countable, since $$\check L$$'s weight always equals its cardinality.

Thus a generalized inverse limit may only be metrizable when $$L$$ is countable or $$f$$ is trivial.

## A few examples

• $$\varprojlim\{I,\gamma,\alpha\}$$ is a copy of the closed long ray of length $$\alpha$$.
• $$\varprojlim\{I,\gamma,I\}$$ is a copy of $$I\times_{lex} I$$.
• Let $$M$$ be a LOTS with $$0$$ and maximum $$1$$. $$\varprojlim\{M,\gamma,L\}$$ is a quotient space of $$\check L\times_{lex}M$$.

## Future work

• What other classic (or new/exotic?!) topological spaces may be expressed as $$\varprojlim\{X,f,L\}$$?
• For uncountable $$L$$, $$\check L$$ can be a $$W$$-space. Can it be Corson compact?
• What minimal criteria guarantee that $$f$$ satisfies condition $$\Gamma$$?
• Can similar techniques be used for a family of bonding relations $$\{f_{\alpha,\beta}:\alpha\lt\beta\in L\}$$?

### References

• 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.