Metrics & Topologies: Do Different Metrics Induce Different Topologies?

[kent davidge]

I was thinking about how different metrics induces different topologies and I was wondering if two different metrics always induces two different topologies.

Does anyone know the answer?

 

[fresh_42]

I was thinking about how different metrics induces different topologies and I was wondering if two different metrics always induces two different topologies.

Does anyone know the answer?

The answer is no. Simple example: If $d(.\; , \;.)$ is a metric, so is $c \cdot d(.\; , \;.)$. In a Euclidean space, these define the same topologies. There is the notion of equivalent metrics.

 

[kent davidge]

The answer is no. Simple example: If $d(.\; , \;.)$ is a metric, so is $c \cdot d(.\; , \;.)$. In a Euclidean space, these define the same topologies. There is the notion of equivalent metrics.

Oh, good counter-example. I was thinking of $\mathbb{S}^1$ and $\mathbb{R}^2$. I would argue that one cannot be made equal to the other because we cannot make the metric of one equal to the metric of the other.

So the argument would be something like: if one cannot make the metrics of two topological spaces equal (or equivalent) then these two topological spaces cannot be made equal, but the converse is not true as by your counter-example.

 

[fresh_42]

If you already start with two different sets $X$, what does it mean to have equal topologies? At least the set should be the same before we start to discuss its structure.

 

[kent davidge]

If you already start with two different sets $X$, what does it mean to have equal topologies?

Oh yea, this is a good point. Let'me give a concrete example of what I'm thinking of. Suppose we have $\mathbb{R}^2$ with the euclidean metric (identity matrix) and the metric induced by the polar coordinate system (components $1$ and $r^2$). These two metrics together with $\mathbb{R}^2$ define two different topological spaces.

Can we tell what metric is being used just by looking at the behaviour of functions?

 

[fresh_42]

Oh yea, this is a good point. Let'me give a concrete example of what I'm thinking of. Suppose we have $\mathbb{R}^2$ with the euclidean metric (identity matrix) and the metric induced by the polar coordinate system (components $1$ and $r^2$). These two metrics together with $\mathbb{R}^2$ define two different topological spaces.

Can we tell what metric is being used just by looking at the behaviour of functions?

They don't. They are simple coordinate systems, which has nothing to do with neither the metric nor the topology. The metric is the Euclidean distance in both cases, no matter how we represent points. A topology is a collection $\mathcal{C}$ of sets with some properties as $X,\emptyset \in \mathcal{C}$ etc. and we call the pair $(X,\mathcal{C})$ a topology. So to have two different topologies, we can either consider different sets $X$ or different collections $\mathcal{C}$. The latter means, we have an element (open set) in one collection which isn't part of the other. How should your metric(s) above define different open sets?

 

[kent davidge]

They are simple coordinate systems

but coordinate systems induces a metric, don't they?

 

[fresh_42]

A coordinate system is a way to identify points in a set, usually a direct product of some other sets. A metric is a distance between two of those points. These are two different concepts. E.g. we can use $e_1=(1,0)$ and $e_2=(0,1)$ as coordinates in $\mathbb{Z}_2 \times \mathbb{Z}_3$ which allows us to identify arbitrary points $(a,b)=a \cdot e_1 + b \cdot e_2$ by how many times our units $e_i$ have to be taken in the corresponding direction. There is no metric at all.

If on the other hand we have a metric, then it has to be said what the distance between two points is. If we describe points by Cartesian coordinates, we get a different formula as if we defined them by polar coordinates. That does change the formula, not the distance. Of course there are also different ways to define distances, e.g. by the discrete metric, and we will again get a different formula.

With a coordinate system we deliberately choose a kind of unit per direction, resp. component. Sometimes these units can also be used to define a metric, and sometimes not. So both are different concepts. A metric is a distance. That's it. The work starts, if we want to define this distance. It leads automatically to the question, how we define the points. If both can be done with the same ruler, fine, but that doesn't have to be.

 

[fresh_42]

but coordinate systems induces a metric, don't they?

No, as my example of finite sets illustrates.

 

[lavinia]

I was thinking about how different metrics induces different topologies and I was wondering if two different metrics always induces two different topologies.

Does anyone know the answer?

The same topological surface given different shapes has different metrics. For instance a sphere and an ellipsoid have the same topology but different shapes. In general if you start with a surface and continuously deform it in space, the topology is preserved while the metric changes. If you follow two points through the deformation their distance will change but open sets will be warped into open sets.