What's the correct relationship between these two spaces?
I think that topological space is a metric space, since the open is defined by a metric such that $d(x, a) < \epsilon$.
[Math] Is Topological Space a Metric Space
general-topologymetric-spacestopological-vector-spaces
Related Question
- [Math] Why is metric space a Hausdorff space but not a topological space
- [Math] what is the relation between measurable space (measure space) and topological space (with a metric)
- Show that if two metric spaces are isometrically isomorphic then the induced topological spaces are homeomorphic.
- Definition of “Topological Equivalence” for metric spaces
Best Answer
Technically a metric space is not a topological space, and a topological space is not a metric space: a metric space is an ordered pair $\langle X,d\rangle$ such that $d$ is a metric on $X$, and a topological space is an ordered pair $\langle X,\tau\rangle$ such that $\tau$ is a topology on $X$. A metric on $X$ is a special kind of function from $X\times X$ to $\Bbb R$, and a topology on $X$ is a special kind of subset of $\wp(X)$, and obviously these cannot be the same thing. Thus, neither class is technically a subclass of the other.
What is true, however, is that every metric $d$ on a set $X$ generates a topology $\tau_d$ on the set: $\tau_d$ is the topology that has as a base $\{B_d(x,\epsilon):x\in X\text{ and }\epsilon>0\}$, where
$$B_d(x,\epsilon)=\{y\in X:d(x,y)<\epsilon\}\;.$$
Thus, people often say, rather sloppily, that every metric space is a topological space.
On the other hand, it is not true that every topology on a set $X$ can be generated by a metric on $X$. A topological space $\langle X,\tau\rangle$ whose topology can be generated by a metric on $X$ is said to be metrizable. Many, many spaces, even quite nice ones, are not metrizable. Thus, it isn’t true that every topological space ‘is’ a metric space, even in the sloppy sense in which every metric space ‘is’ a topological space.