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

  1. 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
     
  2. jcsd
  3. radou

    radou 3,215
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    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.
     
  4. vela

    vela 12,780
    Staff Emeritus
    Science Advisor
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    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_space#Open_and_closed_sets.2C_topology_and_convergence
     
  5. hunt_mat

    hunt_mat 1,613
    Homework Helper

    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.
     
  6. 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.
     
  7. Yes, that is correct. Euclidean space is a concrete example of a metric space, metric space has a topology induced by metric.
     
  8. Deveno

    Deveno 906
    Science Advisor

    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>).
     
  9. This is incorrect. Every closed ball with radius r>0 is contains an open ball with some radius smaller than r, but a closed ball is closed. For a simple counterexample, consider the real numbers equipped with the usual metric. Then [-1,1] contains (-1/2,1/2) which is an open ball, but [-1,1] is closed.

    It would be correct to say that X is open if every element of X is in an open ball that X contains. That is, X is open if for each x in X, there exists an r>0 st B(x,r) is a subset of X.
     
    Last edited: Sep 12, 2013
  10. hunt_mat

    hunt_mat 1,613
    Homework Helper

    Sorry, that should contain an open ball at every point.
     
  11. We shouldn't ask the difference between a metric space and a topological space, indeed it's been mentioned above that sometimes they are the same, and that every metric space is a topological space.

    The source of confusion in the definitions usually has to do with the definition of an open set. In analysis, we define an open set as in my previous post:

    Consider a metric space S. A set X in S is open if for every x in X, there exists some r>0 such that B(x,r) is completely contained in X.

    In a metric space, these open sets have three properties (which I will repeat here):

    1) S and the empty set are open sets.
    2) Any union of open sets is open.
    3) Any finite intersection of open sets is open.

    In a metric space, we can show that these are properties of open sets that follow from the definition of an open set given above. It would be a good exercise to prove each one using the definition. So in a metric space, in order to define what an open set is, we use the notion of distance when we talk about the open ball, namely when we give a radius. An open ball centered at x is the set of all the elements of S that are within that radius of x.

    In a topological space however, you might say we "skip ahead" to the properties. That is, in a topological space, we say that a set is open if it satisfies those three properties. The properties then do not follow from the definition, like in a metric space, but are themselves first principles that we use to define this type of set. So unlike a metric space, we can't prove that open sets satisfy the properties, we take for granted that they do, and we just call the sets that satisfy them "open". This is why we don't require a notion of distance in a topological space, because we don't require balls to define open sets or prove statements about them.

    However, the open sets in any metric space still satisfy the properties. So if T is the collection of open sets in our metric space S, we can say that (S,T) is indeed a topological space.
     
    Last edited: Sep 12, 2013
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