# Metrics which generate topologies

• I
• Lucas SV
In summary: These are very interesting questions. I will definitely be looking at them in more detail.Yes, there is a name for an algebraic structure of things that are closed under addition and multiplication by positive scalars, but not under subtraction. It's called a metric space.
Lucas SV
Given a topological space ##(\chi, \tau)##, do mathematicians study the set of all metric functions ##d: \chi\times\chi \rightarrow [0,\infty)## that generate the topology ##\tau##? Maybe they would endow this set with additional structure too. Are there resources on this?

Thanks

Yes they do. We say the topological space is 'metrizable' if there exists a function of the type you describe that has the properties of a metric and the metric topology that it generates is the same as ##\tau##. Studying metrizability is a key part of studying topology. Metrics provide additional structure because they provide a notion of distance, which does not exist in a bare topological space.

A commonly-used resource on this is Munkres' popular textbook 'Topology, a first course' in which a significant part of the second chapter is devoted to metrizability.

andrewkirk said:
Yes they do. We say the topological space is 'metrizable' if there exists a function of the type you describe that has the properties of a metric and the metric topology that it generates is the same as ##\tau##. Studying metrizability is a key part of studying topology. Metrics provide additional structure because they provide a notion of distance, which does not exist in a bare topological space.

A commonly-used resource on this is Munkres' popular textbook 'Topology, a first course' in which a significant part of the second chapter is devoted to metrizability.

Yes, I've seen the metrizability concept before. But my question is not so much on the existence of a metric function which generates the topology, but given you already know it exists, for instance the Euclidean topology, is there some structure to the set of all such metric functions. For example, it seems this set is closed under addition. One can also ask about the cardinality of such set. Since metrizability focuses on existence, I'm not sure it will help.

Stephen Tashi
Lucas SV said:
but given you already know it exists, for instance the Euclidean topology, is there some structure to the set of all such metric functions.

- "such metric functions" meaning the class of all metric function that are equivalent to the Euclidean metric function.

Is there is a name for an algebraic structure of things that are closed under addition and multiplication by positive scalars, but not under subtraction? (I don't know.)

Suppose ##T: \mathbb{R}^n \rightarrow \mathbb{R}^n## is a 1-1 continuous (in the Euclidean topology) mapping of ##\mathbb{R}^n## onto itself. And let ##d(x,y )## be the Euclidean metric. Can we show ##m(x,y) = d(T(x),T(y))## is a metric that is topologically equivalent to the Euclidean metric?

If that idea holds up then we can ask if the converse holds - if every metric ##m(x,y)## that is topologically equivalent to the Euclidean metric on ##\mathbb{R}^n## can be realized as ##d(T(x),T(y))## for some ##T(x,y)##.

Things will be more interesting if the converse is false. If the converse is true then it looks like the study of "all metrics equivalent to the Euclidean metric" just amounts to the study of all 1-1 continuous mappings of ##\mathbb{R}^n## onto itself.

FactChecker and Lucas SV
Stephen Tashi said:
"such metric functions" meaning the class of all metric function that are equivalent to the Euclidean metric function.
Why a class though? Since such metrics are written as ##d: \mathbb{R}\times\mathbb{R} \rightarrow [0,\infty)##, which can be thought as a subset of ##\mathbb{R} \times \mathbb{R} \times \mathbb{R}##, so the set of all metrics equivalent to the Euclidean metric would be well defined in set theory as a subset of ##\mathcal{P}(\mathbb{R} \times \mathbb{R} \times \mathbb{R})##.

Stephen Tashi said:
Suppose T:Rn→RnT:Rn→RnT: \mathbb{R}^n \rightarrow \mathbb{R}^n is a 1-1 continuous (in the Euclidean topology) mapping of RnRn\mathbb{R}^n onto itself. And let d(x,y)d(x,y)d(x,y ) be the Euclidean metric. Can we show m(x,y)=d(T(x),T(y))m(x,y)=d(T(x),T(y))m(x,y) = d(T(x),T(y)) is a metric that is topologically equivalent to the Euclidean metric?

If that idea holds up then we can ask if the converse holds - if every metric m(x,y)m(x,y)m(x,y) that is topologically equivalent to the Euclidean metric on RnRn\mathbb{R}^n can be realized as d(T(x),T(y))d(T(x),T(y))d(T(x),T(y)) for some T(x,y)T(x,y)T(x,y).

These are very interesting questions. I will definitely be looking at them in more detail.

Lucas SV said:
Why a class though? Since such metrics are written as ##d: \mathbb{R}\times\mathbb{R} \rightarrow [0,\infty)##, which can be thought as a subset of ##\mathbb{R} \times \mathbb{R} \times \mathbb{R}##, so the set of all metrics equivalent to the Euclidean metric would be well defined in set theory as a subset of ##\mathcal{P}(\mathbb{R} \times \mathbb{R} \times \mathbb{R})##.
These are very interesting questions. I will definitely be looking at them in more detail.
Pay special attention to @Stephen Tashi 's point that the set is not closed under subtraction. That doesn't leave much in abstract algebra that can apply. Maybe category theory has some use here, but I don't know enough about that.

## 1. What is the purpose of metrics in generating topologies?

Metric refers to a measure of distance or similarity between objects or data points. In the context of generating topologies, metrics are used to quantify the relationships between elements and determine the structure of the resulting topology.

## 2. How do metrics affect the accuracy of generated topologies?

The choice of metrics can greatly impact the accuracy of generated topologies. Metrics that are not appropriate for the data being analyzed can lead to incorrect or misleading topologies. It is important to carefully select metrics that are relevant to the data and the desired outcomes.

## 3. Can metrics be customized for different types of data?

Yes, metrics can be customized or developed specifically for different types of data, depending on the nature of the data and the desired outcomes. For example, metrics used in biological data analysis may differ from those used in social network analysis.

## 4. Are there any limitations to using metrics for generating topologies?

While metrics are useful tools for generating topologies, they do have limitations. Metrics may not accurately capture complex relationships between elements, and different metrics may lead to different topologies. Additionally, the interpretation of metrics and resulting topologies may be subjective and dependent on the goals of the analysis.

## 5. How can one evaluate the effectiveness of metrics in generating topologies?

The effectiveness of metrics in generating topologies can be evaluated by comparing the resulting topologies to known or expected structures, or by assessing the consistency and stability of the topologies generated using different metrics. It is also important to consider the relevance and interpretability of the metrics in relation to the data and research goals.

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