# Dictionary Order Topology on ##\mathbb{R}^2## Metrizable?

• Bashyboy
In summary, the conversation discusses finding a metric on ##\mathbb{R}^2## that induces the dictionary order topology on the plane. The speaker is trying to define a metric that makes the corresponding ##\epsilon##-balls vertical strips, but is unsure if their proposed function is valid. They suggest using ##\infty## as a placeholder, but later realize that limiting the balls to a radius less than ##1## effectively acts as ##\infty##. The speaker also expresses interest in knowing if their initial function is a valid metric.
Bashyboy

## Homework Statement

I am trying to show that there exists a metric on ##\mathbb{R}^2## that induces the dictionary order topology on the plane.

## The Attempt at a Solution

If I recall correctly, vertical intervals in the plane form basis elements for the dictionary order topology. With this in mind, I am trying to define a metric such that the corresponding ##\epsilon##-balls are vertical strips. The only function that I could come up with is

$$d((x,y),(w,z)) = \begin{cases} \infty ,& x \neq w \\ |y-z| ,& x=w \\ \end{cases}$$

Based on my work, it seems that this is a well-defined metric. However, somehow I feel that I am cheating by using ##\infty##. Is this in fact a valid metric?

Okay, I think I may have a way to "fix" this, though I would still be interested in knowing if the above defines a valid metric. When dealing with a metric induced topology whose basis elements are ##\epsilon##-balls, it suffices to look at those ##\epsilon##-balls with radius less than ##1##. Now, if we replace ##\infty## in the above function and confine ourselves to dealing with ##\epsilon##-balls of radius less than ##1## , then the metric will restrict the balls from containing points "off" the vertical line.

More precisely, if ##(a,b) \in B((x,y), \epsilon)## and ##a = x##, then ##d((x,y),(a,b)) = 1 > \epsilon##, a contradiction. So, if ##(a,b)## is a point in the ball, then necessarily ##a \neq x## and therefore ##d((x,y),(a,b)) = |y-a| < \epsilon## or ##a \in (y-\epsilon,y+\epsilon)##. This, of course, suggests that ##B((x,y),\epsilon) = \{x\} \times (y-\epsilon,y+\epsilon)##, which is a basis element of the dictionary order topology. In this case, ##1## effectively acts as ##\infty##.

As I mentioned above, I would still be interested in knowing whether the function in my first post defines a valid metric.

## 1. What is the Dictionary Order Topology on ##\mathbb{R}^2## Metrizable?

The Dictionary Order Topology on ##\mathbb{R}^2## is a specific topology on the Cartesian product of the real line with itself. It is defined by taking the Cartesian product of ##\mathbb{R}^2## with itself and ordering the resulting elements in a dictionary-like manner. This topology is metrizable, meaning it can be described by a metric space, and is often used to study ordered sets and their properties.

## 2. How is the Dictionary Order Topology on ##\mathbb{R}^2## different from the standard topology on ##\mathbb{R}^2##?

While the standard topology on ##\mathbb{R}^2## is defined using open balls, the Dictionary Order Topology on ##\mathbb{R}^2## is defined using open intervals. This means that the open sets in the Dictionary Order Topology may look different from those in the standard topology, and their properties may also differ.

## 3. What are the advantages of using the Dictionary Order Topology on ##\mathbb{R}^2## in research?

The Dictionary Order Topology on ##\mathbb{R}^2## has many useful properties that make it a valuable tool in research. It is a metrizable space, meaning it is well-behaved and can be described using a metric function. It is also a Hausdorff space, meaning any two distinct points have non-overlapping neighborhoods. Additionally, it is a locally compact space, meaning every point has a compact neighborhood, making it useful for studying convergence and continuity properties.

## 4. Can the Dictionary Order Topology on ##\mathbb{R}^2## be extended to higher dimensions?

Yes, the Dictionary Order Topology can be extended to any finite number of dimensions. The Cartesian product of n copies of ##\mathbb{R}##, denoted by ##\mathbb{R}^n##, can be equipped with the Dictionary Order Topology in the same way as ##\mathbb{R}^2##, by ordering the elements using a dictionary-like ordering. This extended topology also shares many of the useful properties of the Dictionary Order Topology on ##\mathbb{R}^2##.

## 5. How is the Dictionary Order Topology on ##\mathbb{R}^2## related to other common topologies?

The Dictionary Order Topology on ##\mathbb{R}^2## is closely related to the standard topology on ##\mathbb{R}^2##, as well as the order topology on ##\mathbb{R}^2##. In fact, the Dictionary Order Topology can be seen as a combination of these two topologies. It is also related to the product topology, as it is a special case of the product topology on ##\mathbb{R}^2##. Additionally, the Dictionary Order Topology on ##\mathbb{R}^2## can be used to construct other topological spaces, such as the Sorgenfrey line.

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