Topological space, Euclidean space, and metric space: what are the difference?

michonamona

Hello my friends!

My textbook has the following statement in one of its chapters:

Chapter 8:Topology of R^n
If you want a more abstract introduction to the topology of Euclidean spaces, skip the rest of this chapter and the next, and begin Chapter 10 now.

Chapter 10 covers topological ideas in a metric space setting. I understand what a metric space is (a set of points over which there is defined a distance function that satisfy three special properties), but I don't understand how the three concepts mentioned in the title are related.

1. Are every euclidean space a metric space? is the converse true? are all metric spaces a euclidean space? why not?

2. What is the meaning of "topology of Euclidean spaces"? How is topology related to metric spaces?


I appreciate you any input you guys can contribute. Thanks.

M

radou

Every Euclidean space is a metric space equipped with the standard Euclidean metric. But every metric space is not an Euclidean space.

Topology is related to metric spaces because every metric space is a topological space, with the topology induced from the given metric.

vela

A space is Euclidean because distances in that space are defined by Euclidean metric. If a metric space has a different metric, it obviously can't be a Euclidean space.

When you have a metric space, you have the concept of an open ball, which in turn leads to the concept of open sets on the space. These are the open sets that make it a topological space.

http://en.wikipedia.org/wiki/Metric_...nd_convergence

hunt_mat

A metric space is a pair (X,d) which consist of a set X and a map [tex]d:X\times X\rightarrow\mathcal{R}[/tex] which is called the metric (think of it as a function which measures distance between two points in the set X). The metric satisfies the following conditions: [tex]1)d(x,x)=0\quad 2)d(x,y)=d(y,x)\quad 3)d(x,y)\leq d(x,z)+d(z,y)[/tex].

The open ball at a point x is defined as [tex]B(x,\varepsilon ):=\{ y\in X|d(x,y)<\varepsilon \}[/tex]. An open set is any set in X which contains an open ball. Euclidean space has the metric [tex]d(x,y)=\sqrt{(x_{1}-y_{1})^{2}+\cdots +(x_{n}-y_{n})^{2}}[/tex]
Open sets satisfy the following conditions:
1) The empty set and X are both open
2) An arbitrary union of open sets is open
3) Any finite intersection of open sets is open
The above three axioms define a topology on X. A Topological space T, is a collection of sets which are called open and satisfy the above three axioms. In general topological spaces do not have metrics.

michonamona

Thank you for your replies.

Would it be safe to make the following generalization?

topological space--->metric space---->euclidean space

This means that every euclidean space is a metric space and every metric space is a topological space. By transitivity, every euclidean space is a topological space.

losiu99

Yes, that is correct. Euclidean space is a concrete example of a metric space, metric space has a topology induced by metric.

Deveno

yes. certain spatial properties of euclidean space are abstracted to get the notion of a topological space.

metric spaces are in-between the two, they are a special kind of topological space, but there are several possible metrics on a given set, including R^n. of these, only one is the standard euclidean metric on R^n: d(x,y) = √(<x - y,x - y>).